The INK is a rogue faction of the KND.The members of INK were once a sector of the KND's French division, but ended up deserting due to some unknown transgression. To avoid the obligatory mind wipe that comes with leaving, they went to Pinkerton Academy, the location of which is unknown to everyone except for a certain few (the students and the teachers). When they found out that there was adult tyranny happening at the school, they were more than willing to take up the fight against it and created their own team to do so.
Ms. Macbeth is a descendant of Scottish CatholicsSome of her ancestors probably emigrated to France after the Scottish Reformation of 1560. At the very least, it would explain why she has a Scottish surname even though she (presumably) lives in France.