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.