# Does an equivariant weak equivalence induce weak equivalences on all orbits?

This question arose from another, which was not well formulated and completely answered by this MO thread as pointed out by the user roman.

Let $G$ be a discrete group, $X$ and $Y$ be $G$-spaces (and compactly generated, weak Hausdorff) and $f:X\to Y$ a $G$-equivariant map which is an equivariant weak equivalence, i.e. for all subgroups $H$ of $G$ the induced map $f^H:X^H\to Y^H$ on fix-points is an ordinary weak equivalence.

Does it follow that for all subgroups $H$ of $G$ the map $f/H:X/H\to Y/H$ on quotient spaces is an ordinary weak equivalence? A quotient space is the coequalizer of the two maps $H\times X\to X$, the action and the projection.