free $G$-action on a topological space

Question

Does existence of a free finite $G$-action on a topological space invariant under homotopy? In other words, let $X$ be a free $G$ topological space which is homotopic with a topological space $Y$. Does this assumption imply of the existence of a free action of $G$ on Y? Additionally, assume $X$ and $Y$ are finite CW-complex and $G$ is a finite group.