Let $(\tilde{X},p)$ be a universal covering space of $X$. We know that if $G$ acts properly disctinously on $\tilde{X}$, then $\tilde{X}$ is a covering space of $\tilde{X}/G$ and $\pi_1(\tilde{X}/G)=G$; in particular, $X= \tilde{X}/ \pi_1(X)$.
Let $G$ be a subgroup of $\pi_1(X)$. There is a canonical map $q : \tilde{X}/G \to \tilde{X}/\pi_1(X)=X$ corresponding to the injection $G \to \pi_1(X)$.
My question: Is $(\tilde{X}/G,q)$ a covering space of $X$?
By my preliminary remark, $(\tilde{X},\pi)$ is a covering space of $\tilde{X}/G$ with the canonical surjection $\pi : \tilde{X} \to \tilde{X}/G$. So we have the following commutative diagram:
$$\begin{array}{ccc} \tilde{X}/G & \rightarrow^{q} & X \\ \uparrow{\pi} & \nearrow{p} & \\ \tilde{X} & & \end{array}$$
My main problem is that if $U \subset X$ then the connected components of $q^{-1}(U)$ don't have necessarly the same behaviour.