Am I allowed to send my group a fake report?
Are you kidding, a fake one? Why would you send them a report that is fake?
So the committee will have to expel me, not them
It's horrible to lie like that. Your group is going to hate you for that.
They will understand I had to do it since the committee failed me.
Lying, no matter who you want to protect, will be something you need to explain. Just do it the honest way and see what happens, maybe you can work things out.