I am trying to prove that $X/H \rightarrow X/G$ is a normal covering if and only if $H$ is a normal subgroup of $G$.
In my particular case, $X$ is the universal covering of $X/G$ and $G$ is the deck transformation group of $X$ over $X/G$. I do not want to deal with the identification between deck transformation groups and the fundamental group where the proof is given in a lot of sources.
I found this answer that deals with a very similar problem to mine in item 3.
Group action and covering spaces
However, I can not see why for a deck transformation $\phi \in G= Deck(X \rightarrow X/G)$ there is an induced deck transformation $\tilde{\phi} \in Deck(X/H \rightarrow X/G)$. My main concern is why the map $\phi$ preserves $H$-equivalence classes and gives a well defined map $X/H \rightarrow X/G$.
