Let $X$ be a $CW$-complex and let $G$ be a group acting "cellularly" on $X$, that is, for each $g\in G$, the induced homeomorphism $\phi_g:X\to X$ takes cells to cells.
Also assume that the action is free.
Then it seems to me that the orbit space $X/G$ also becomes a CW-complex.
Question. Are there any results knows which relate the homologies of $X/G$ with the homology of $X$, taking into account the action of $G$?
In general, are there any results known which relate the homologies of the orbit space of a $G$-space $X$, where now $X$ is just any topological space and not necesarily a CW-complex.