Basically, a group who separates from a larger organization - holding the main beliefs or goals of the original, but are wholly or mostly evil.
There are two common ways of applying this trope:
The Doylist Angle: The writer doesn't want to paint some organization, nation or religion black. It may be a real-world institution, or an institution from this fictional world which was previously described as at least semi-decent. But the author still wants villains belonging to it, so they invent a renegade group.
The Watsonian Angle: A faction that operates without authorization, and their methodology differs from the main body. Often they are extremists: escalating a conflict, willing to break the rules, far more infamous than their mother alignment - but this isn't always the case.
Note that it doesn't apply to out and out turncoats: the faction must still keep the general ideas of the original; only with less benevolent policies. If the factions are religious in nature, related to The Heretic. Can result in Enemy Civil War.
Blue Cosmos, an anti-Coordinator terrorist group and movement within the Earth Alliance. Many in the Alliance don't share such extremist views. Sadly, Blue Cosmos ends up gaining enough influence to control the policies of the government.
The Clyne Faction. Originally the Moderates within ZAFT, they broke off when the Zala extremists came to power. Subverting the trope, they're outright good guys. The Faction managed to form a underground resistance, steal key equipment, and join up with the heroes.
The Treize Faction in Gundam Wing. A splinter group of OZ / the Romafeller Foundation, the Faction opposed the introduction of automated mecha. An inverse of the trope, the splintering had made Foundation come off as increasingly villainous.
La Eden, a violent right-wing political group within the AEU, though not supported by the Superpower. The group commits a series of terror attacks to stop the protagonists. Celestial being responds by simultaneously curbstomping the entire faction.
Team Trinity claims to be part of Celestial Being, sharing the similar technology and goal of eliminating warfare. However, the Trinities are much more ruthless - attacking without provocation nor mercy. Eventually, the two groups come into conflict.
The Innovators were originally part of Celestial Being. But thanks to the events of Season 1, the two groups splinter, with the Innovators causing most of the problems for the next season.
The ESF Coup detat Faction. Splintering off from the Federation military, the Coup seeks to expose the A-Laws' atrocities. Like the Trieze Faction, their going "renegade" makes the Earth Sphere Federation come off as increasingly antagonistic.
The Titans in Mobile Suit Zeta Gundam. Though officially part of the Earth Federation serving as a special security force, the group was used by a corrupt politician to gain control of the Earth Sphere. Their actions were zealous, which including gassing an entire colony. Eventually they went full renegade after they were exposed by the AEUG.
The sinister Beylin Fist faction in Metal Fight Beyblade. The group had splintered from China's Beylin Temple training school.
In Digimon Tamers, the protagonists thought that since the Devas were serving one Digimon Sovereign, then all four might be against them. It turns out Zhuqiaomon, the Sovereign who the Devas serve, is acting on his own. He and the other three leading Digimon couldn't agree on a single defense strategy against the D-Reaper.
WILLE of Rebuild of Evangelion, consisting of former NERV agents who oppose the progenitor agency. An inversion of the trope, WILLE goals are benevolent compared to NERV.
Nextwave split itself off from H.A.T.E. when they learned that HATE was actually owned by the Beyond Corporation, which was a cover group for S.I.L.E.N.T., their ostensible nemesis. Nextwave tries to do good when they're not on the run from their employers.
The Sand Blast City Freedom Fighters in Archie Comics' Sonic the Hedgehog. Unlike the other Freedom Fighters, the Sand Blasters are ruthless and their actions questionable. It was because of their abusive treatment that the deroboticized Robians joined the Eggman Empire. Yes, it got so bad that the former slaves entered into an alliance with their former master.
In Marvel Comics, Hydra has had several splinter groups: AIM, the Hand, and the Secret Empire among others.
The Crimson Jihad in True Lies is a splinter group that broke off of a larger Islamic terrorist organization, apparently because their leader Salim Abu Aziz didn't think the bigger group was crazy enough.
General Chan Lu from the remarkably silly Battle Beneath The Earth is an example of the Renegade Chinese version.
In Star Trek IV: The Voyage Home, the Klingon Ambassador tries painting Captain Kirk as a "renegade and terrorist" due to the events of the previous film. The likely reason why the Ambassador doesn't claim the entire Federation is on a conspiracy to "annihilate the Klingon species" is political tact.
The crew of the Narada in the 2009 Star Trek. When Captain Pike confronts the Narada, he states that the Federation and the Romulan Star Empire are not at war with each other. However, Nero replies that his crew do not represent the Empire, as they "stand apart".
Fëanor and his sons and followers in The Silmarillion are more or less a Renegade Faction of the High Elves.
In the novel Patriot Games and it's film adaption, the Ulster Liberation Army is a Marxist splinter faction of the Provisional IRA. Naturally, the ULA is more ruthless than the group they left.
In the Frontier Magic series, the mainstream Rationalists simply believe that not being dependent on magic is a good idea; but an isolated settlement of fanatical Rationalists ends up condemning magic as evil.
The Faithful of the Church of Humanity Unchained, who settled Masada in the Honor Harrington series. Originally part of the Church of Humanity Unchained on the planet Grayson, they broke from the main church over theological disputes. In contrast to the Grayson church, the Masadans are such extremists that they've rejected all of the New Testament.
The Second Imperium, the Imperial faction from the Star Wars: Young Jedi Knights book series was retconned to be one of these, because another series, Hand of Thrawn, established that a peace treaty was brokered between the Republic and the mainstream Imperial Remnant under the leadership of the relatively nice mustachioed Admiral Gilad Pellaeon.
Section 31 of Star Trek: Deep Space Nine is this for The Federation. A covert and officially nonexistent group, the Section deals with any threats to the Federation, even if it has to violate it's principles.
Another Star Trek example is the Maquis - a paramilitary group that oppose the Cardassians. Originally consisting former Federation citizens, the group also had Starfleet officers aiding them illegally.
The rogue NID of Stargate SG-1 was this for Earth. While the regular NID was rather shady, the renegade group outright criminal.
Eventually, the rogue NID finally splintered off into The Trust once their wealthy backers were arrested. Freed from any government restrictions, The Trust was even more violent - conducting chemical attacks and assassination. When their operatives were driven away from Earth by the SGC, the group managed to get themselves captured and implanted with Goa'uld. Ironically, Ba'al and Athena ended up in charge of The Trust, and headed several major corporations.
Homeguard in Babylon 5. The group was a xenophobic paramilitary faction within the Earth Alliance, committing attacks on aliens.
Alias: SD-6 worked as a sometimes-splinter group of the Alliance of Twelve, since its head Arvin Sloane had his own agenda apart from the Alliance.
An episode of NCIS features a group called the "MAH" (Military At Home), who believe that America should focus on fighting crime and social problems, instead of policing the world. A group of MAH decides to take it a step further, plotting to destroy a communications tower to make a point, with no intention of harming anyone. However, the renegade group also had a splinter group who decide to attack people as well.
The Word of Blake in BattleTech, which broke off from ComStar. When the mother organization reformed to lessen focus on it's mystical aspects, the conservatives broke off. The Word eventually started an all-out war that affected in entire Inner Sphere.
The eponymous Delta Green. Once working for the government, it illegally continues its operations to defend America from the Mythos.
2nd Edition Dungeons & Dragons supplement The Complete Druid's Handbook. The Shadow Circle is a secret society of druids within the larger druidic order. It sees Nature as a hostile, cleansing force that ensures the survival of the fittest and thinks that civilization has weakened humanity and the demihuman races. They support barbarians and think that people should go back to nature. They use evil and vicious tactics to carry out their beliefs.
This basically happened to Team Plasma in Pokémon Black 2 and White 2. N's Plasma just wanted people to stop using pokeballs, though they're still on the 'release your Pokemon' kick. Ghetsis' group wants to take over Unova.
The Brotherhood Outcast faction in Fallout 3 split off from Elder Lyons faction after he decided to actively involve himself in helping the people of the Capital Wasteland. The Outcasts consider themselves the "true Brotherhood" and believe that Lyons and his group have strayed from their original mission.
One mission of Command & Conquer: Generals has a renegade Chinese army (allied with the Middle-Eastern terrorist faction which is the villain of the game) as one of the enemies. In this game, China is one of the "good" factions.
The NSA in Perfect Dark. The director of the agency, Trent Easton, is in league with a shadowy conspiracy. And unlike today's NSA, Easton seems to have his own personal army. In order to accomplish their goals, Easton plans to replace the President with a clone as part of a larger scheme.
Mass Effect 2 reveals that the geth you spent most of the first game fighting are in fact a splinter faction considered heretics by the mainstream geth population, who call themselves the True Geth. The True Geth believe that they and all other creatures should be free to choose their own fates, whereas the heretics serve and obey the Reapers.
Guild Wars 2 has this for every player race: Humans have bandits, rogues, cutthroats, and other violent criminals. Charr have the dogmatic Flame Legion. Asura compete with amoral Mad Scientist types. Sylvari fight the alien Nightmare Court. Nords struggle against the vicious Sons of Svanir.
In Command & Conquer: Tiberium series, The Brotherhood of Nod has a chronic issue with these. Various factions of the Brotherhood end up fighting with each other for power; Kane is the only one to really unite them all. Here are a few notable examples:
The Black Hand, led by Anton Slavik. They go renegade when Hassan (Nod's leader and GDI puppet) tries to execute Slavik.
CABAL, after manipulating both sides to accomplish it's goals. It was so much of a threat that both GDI and NOD work together to eliminate it.
The Marcion-led Black Hand, which was but one of the many splinter factions after CABAL's defeat and Slavik's death. Believing the Brotherhood strayed from it's "pure" path, Marcion took the Black Hand into exile, and reformed the group into a religious order.
The Nod Separatists, who opposed Kane's alliance with GDI. They saw it as a betrayal to their own core beliefs.
GDI tends to be a rather stable bunch, but in Tiberian Twilight one GDI army goes rouge over the alliance with Kane, which leads to infighting within the organization.
Skullgirls lore has The Anti-Skullgirl Labs. They conducted research in order to understand the Skull Heart and develop weapons to destroy the Skullgirl it periodically creates. Unfortunately, they weren't exactly known for the humane treatment of their test subjects. After his wife became a Skullgirl and was subsequently killed, the King had a change of heart and decommissioned the Labs, but latter disappeared under mysterious circumstances. Several members of the ASG Labs took advantage of this and moved to restart the experiments, this time without official supervision and with even shadier laboratory practices.
Finally, one sect, Lab Zero, broke off entirely. The members of Lab Zero operate in complete secrecy, even from the other rogue Labs. Their methods are also much, muchmoreheinous than the others.
It's implied that the restarted Labs may have reformed to be more ethical after Princess Parasoul came into power, making the situation an at least partially subverted example of the trope.
After the Covenant disbanded after their defeat in Halo 3, the Sangheili who did not ally with humanity broke away to form their own Covenant remnant, which are fought by Master Chief in Halo 4. They are unofficially called "the Storm".
In Galactic Civilizations II, mirroring (and probably a deliberate Shout-Out to) the Star Control example, the Korath Clan, which believes that other races should be exterminated outright, rebels against the villainous Drengin Empire.
The human-supremacist terrorist organization Cerberus in the Mass Effect series started out as the black-ops division of the Human Systems Alliance.
The Majestic 12 from Deus Ex are that part of the Illuminati who shed "ethical inflexibility".
Islamic extremism in Real Life. Islam as such does not command anyone to drop planes on trade centers.
Five hats means that five tropers think it is ready to publish.
You are saying that you think this draft is ready to be published. That means the description is not ambiguous,
it doesn't duplicate an existing trope, there are at least three examples, and the title makes sense.
Is that what you meant to do?
You are saying this draft has a ready-to-publish hat it does not deserve and you are taking it back.