Diffeomorphims compatible with an action.

Question

Hello I'm reading the following thread

How to write a diffeomorphism between an open set of the real projective plane $P^2(\Bbb{R})$ and the Möbius strip

But in the same I find the following afirmation

is a diffeomorphism which is compatible with the action of G. It therefore induces a diffeomorphism

But I don't know where find this proposition or how to prove the same. Could anyone help me ?