I am trying to understand the following statement on Allen Hatcher's algebraic-topology textbook in the Section of Local Coefficient:
"A bundle of groups $E\rightarrow X$ with fiber $G$ is equivalent to an action of $\pi$ on $G$. In more explicit terms this means that if $\tilde{X}$ is the universal cover of X, then $E$ is identifiable with the quotient of $\tilde{X}\times G$ by the diagonal action of $\pi$, $\gamma(\tilde{x},g)=(\gamma\tilde{x},\gamma g)$ where the action in the first coordinate is by deck transformations of $\tilde{X}$."
Here "$\pi$" is $\pi_1(X)$, $G$ as a typical fiber is given the discrete topology, and the action of $\pi$ on each fiber $G$ is given by permutations induced by the lift of loop $\gamma$. Besides the proof of the identification above, I also want to ask: why do such permutations belong to the automorphism of $G$?