You say get a caretaker but it isn't that easy, who do we turn to?
Maresca essentially resigned, and Amorim orchestrated an exit by blowing up. Frank is playing the game and wants to stay so, as long as Frank doesn't lose 3 or 4 on the trot, he will probably stay until the summer when it's easy to hire a proper manager.
I'd sack Frank now, as the writing is on the wall, but I just think the board will leave it as long as possible. Also, because any transfer targets we want to sign this window, we need to give off some semblance of stability and a plan.
That said, I do think they would be making a mistake. The season could be salvagable as it's tight, if we sacked Frank now, but that only works if we have a decent replacement to come in.