TITLE: Is there any relation about rational homology of X and X/G
QUESTION [6 upvotes]: If we know the rational homology of $X$ is $0,$ can we get some information about the rational homology of $X/G,$ where $G$ is a finite group? Thank you very much for the answers!

REPLY [7 votes]: When $G$ is finite, the rational cohomology of $X/G$ is the fixed point set $H^*(X;\mathbb{Q})^G$.  This is proven in Grothendieck's Tohoku paper (Theorem 5.3.1 and the Corollary to Proposition 5.3.2).
So if the rational cohomology of $X$ is trivial, the same is true for $X/G$. And rationally the cohomology and homology are isomorphic.
For paracompact Hausdorff spaces, these cohomology groups can be taken to be the Cech cohomology groups.  Note that if $X$ is homotopy equivalent to a CW complex, then Cech cohomology agrees with singular cohomology.  You might also want to look at Oscar Randall-Williams comments here: https://mathoverflow.net/questions/18898/grothendiecks-tohoku-paper-and-combinatorial-topology/30015#30015.