Let $Y$ be a connected CW-complex and $G$ a group acting freely on $Y$ by permuting the cells. We assume the action on $Y$ to be cocompact so that $X = Y/G$ is a finite CW-complex. Then how to see $G$ is a factor group of the finitely generated fundamental group $\pi_1(Y/G)$?
Actually,I think it requires $Y$ to be a covering space of $X$, but I don't think it's true just by free action.