# Homotopy orbit spaces of representation spheres

Let $G$ be a finite group and $V$ be finite-dimensional real representation of $G$. Write $S^V$ for the one-point compactification of $V$, with induced $G$-action, viewed as a pointed space, and consider the homotopy orbit space $(S^V)_{hG}$. For instance, if $V$ is the regular representation of $G=\mathbb{Z}/2$, $(S^V)_{hG}$ works out to be the suspension of $B\mathbb{Z}/2$. Is there some kind of general description of this space, presumably built out of classifying spaces of subgroups of $G$? What if I only care about the stable homotopy type? I'm most interested in the permutation representations of the symmetric groups.

-