So this may just be me being paranoid, but a few things stood out to me about this new girl. First, she is trying to find out why Sayu doesn't go to high school, while also not sharing her own reason for not being in school (Heck from her excitement it sounds like she didn't finish middle school), she then tries to figure out where Sayu lives and decides to invite herself over immediately without any idea of her home life, not bringing Sayu to her place instead. Maybe I'm jumping at ghosts but I just have this hunch that this girl may be involved in something bad and is trying to drag Sayu in, so she's getting friendly, learning where she lives, and integrating with Sayu's life to ensure she can't escape once she gets her involved.
But my paranoia picks up on a lot of things that turn out to be nothing so feel free to disregard.