I still like Rex and I think he may be a good head coach someday. There is a learning curve and we all remember how Belichick was in Cleveland.
But when you bring in a new GM, you need to trust his decision making and he must be the one that chooses the head coach. If he wants to retain Rex, then fine, but if not, goodbye.
And even if he is retained, the GM should be making all personnel decisions. Not Rex. He has proven that he isn't very good at it.