If the mom asks Aoi to stay with the son, then it would seem strange to make Akira transfer elsewhere after a short while.
Considering Aoi's personality, I wonder if she can withstand the pressure of this lie. Akira has purposefully acted as if they can't become a couple, due to his impending transfer. But does Aoi understand that's the one and only reason? She might believe Akira doesn't see her as romantically interesting at all. So, now pretending to be a couple could, in her opinion, be uncomfortable for Akira. Aoi definitely wouldn't want to cause any trouble for Akira. Add to that she might believe the lie could hurt Akira's relations with his parents.