Not much of their backgrounds were explained either. Mutsuki doesn't really have much of a backstory, but Kayoko's has some implications.
Mutsuki: Aru's childhood friend, has known her since at least middle school before Aru's obsession with wanting to be a cool outlaw (back when Aru still wore glasses)
Kayoko: Has some sort of connection with Ako from the prefect team, or at least they know each other, and they hold some kind of contempt for each other. Dislikes the student council of Gehenna too I think. Other people think she may have been a higher-up in Gehenna but nothing is confirmed, it's just kept mysterious for now. Its an interesting theory given Aru's Problem 68 group are known as troublemakers.