Idol likely didn't get to the same school after discovering someone she knew had died and her family situation that made clear it was suicide.
Even if they weren't in that school is likely she knew that FMC would get into it, maybe next we'll discover she wanted to make peace, and had no reason to get in anymore.
Whatever it is the reason for MC not knowing the idol had gone into their school, no matter how detached he was after the funeral he should have at least heard of it, is most likely because she never was in the institute.