Let $G$ be a finite group, and $X$ a free topological $G$-space which admits a CW-structure. Is there a CW-structure on $X$ that compatible with its $G$-action, i.e, a cell structure that turns $X$ to a G-CW-complex?
Note, the n-sphere $\mathbb{S}^n$ with the antipodal action, i.e., $x\to -x$ is an example of free $\mathbb{Z}_2$-space. The cell structure of $\mathbb{S}^n$ with just two cells does not respect the action while the usual cell structure which has two cells in each dimension does.
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