TITLE: Künneth theorem for G-space
QUESTION [5 upvotes]: Let $X$ and $Y$ be a right $G$-space and a left $G$ space, respectively, where $G=H \rtimes K$, $H$ a finite group and $K$ a compact Lie group.
Moreover, suppose that the $G$-action on $X$ is free. 
Denote $H_*(-)$ the singular homology with coefficient in a field.
Is $H_*(X \times_G Y )$ isomorphic to $H_*(X) \otimes_{H_*(G)} H_*(Y)$?
Is it possible to prove it without using spectral sequences?

REPLY [13 votes]: This is very untrue. Taking $X=EG$, a contractible space with a free $G$-action, and $Y$ to be a point we have $H_*(X\times_G Y)=H_*(BG)$, which may be non-trivial in arbitrarily high degrees, while $H_*(X)\otimes_{H_*(G)} H_*(Y)$ is the homology of a point.