If $(\tilde{X},p)$ is a Covering Space of $X$ and $Aut(\tilde{X},p)$ is the group of the Automorphism in $\tilde{X}$, what is the correct definition of $\tilde{X}/Aut(\tilde{X},p)$?
I know that $p:\tilde{X}\rightarrow X$ is an open map, so we can see $X$ as a quotient space of $\tilde{X}$, and this recall that the automorphism group permutes the points of the set $p^{-1}(x)$ among themselves.
At this point my question is:
$$\tilde{X}/Aut(\tilde{X},p)=[\tilde{x}], \ \tilde{x}\in\tilde{X}$$ $$with \ [\tilde{x}]=\{\varphi(\tilde{x})\mid\varphi\in Aut(\tilde{X},p)\}?$$
And if this definition is correct, is correct to say that $\tilde{X}/Aut(\tilde{X},p)\simeq X$ if and only if $Aut(\tilde{X},p)$ operates transitively on $p^{-1}(x)$ too?