Given a covering map $p:Y \to X$, Let's assume we have all basic path-connected and locally path-connected assumptions. Let $G(Y)$ be its Deck transformation. Since we didn't assume $p:Y \to X$ is normal covering, we don't have $Y/G(Y)=X$. But we know (it is not hard to show, see Does Quotienting out by the Deck Transformations Give a Covering?) $p:Y \to X$ factors through two covering spaces as follows: $\require{AMScd}$ \begin{CD} Y @>q>> Y/G(Y) \\ @VpVV @VV\tilde{p}V \\ X @>id>> X \end{CD}
It's not hard to show that $q$ is a normal covering map and the covering space $Y/G(Y) \to X$ is corresponding to $\tilde{p}_*(\pi_1(Y/G(Y)))\subset \pi_1(X)$ which is the normalizer subgroup of $p_*(\pi_1(Y)) \subset \pi_1(X)$. We also have $p_*(\pi_1(Y))=\tilde{p}_*(q_*(\pi_1(Y)))$.
My question is what is $\pi_1(Y/G(Y))$ and $q_*(\pi_1(Y))$? Are they equal to their images seperately in $\pi_1(X)$ by $\tilde{p}_*$, (maybe $\tilde{p}_*$ is injective)?
Thank you in advance for the help!