Suppose I have a topologycal space $X$, which admits a $CW-$structure, and a free $G-$action on $X$, where $G$ is a finite group. I am wondering whether I can choose a $CW-$structure on $X$ such that G acts freely on the cells of $X$.
I know similar questions have been asked before (e.g. here ), and I understand that, since $X \to X/G$ is a cover map, it is enough to give a $CW-$structure on $X/G$ and then lift it. But is it always true that the cover induces a $CW-$structure on $X/G$?
If this isn't true in general, I am particularly interested in the case where X is a sphere $S^n$:
Given a free action of a finite group on the $n-$sphere $S^n$, is it always possible to give it a $CW-$complex structure such that the action is free cellular?
