So, I was going through Hatcher's argument on how covering spaces can be classified by their association with permutation groups, and was able to understand everything until here:
1) How exactly is it evident that $L_{\gamma}(h(\tilde{x_{0}})) = h(L_{\gamma}(\tilde{x_{0}}))?$ Is it something to do with the fact that both $L_{\gamma}$ and $h$ are bijections?
2)Why is the relationship above "what we would naturally call an isomorphism on actions"?
3)How does the isomorphism $h:F_{1} \rightarrow F_{2}$ induce an isomorphism $h:\tilde{X_{1}} \rightarrow \tilde{X_{2}}$?
4) Finally, why is the identification in the last paragraph of conjugacy groups the relation we use? How did Hatcher reach this conclusion from the preceeding paragraph?