Let $p:Y\to X$ be a covering map and $G$ be the group of deck transformations of this cover. Assume for niceness that $X$ admits a universal cover.
It is well-known that the natural projection $\pi:Y\to Y/G$ is a covering.
But the map $p$ factors through $\pi$ to induce a map $\bar p:Y/G\to X$.
Is this map also a covering?