Let $BO_n$ and $BG_n$ be the classifying spaces for rank $n$ vector bundles and for spherical fibrations with fiber $S^{n-1}$, respectively, and let $G_n/O_n$ be the homotopy fiber of $BO_n\to BG_n$.

There is an interesting old fact, that the stabilization map $G_n/O_n\to G_{n+1}/O_{n+1}$ is $(2n-4)$-connected, about twice as good as the corresponding maps $BO_n\to BO_{n+1}$ and $BG_n\to BG_{n+1}$. Equivalently, the map $G_n/O_n\to G/O$ is $(2n-4)$-connected. I was wondering about the history of this result.

But what I was really wondering about was the following equivariant analogue. Let $\Gamma$ be a finite group. For a real representation $V$ of $\Gamma$, let $O(V)^\Gamma$ be the group of linear (isometric) automorphisms respecting the $\Gamma$-action and let $G(V)^\Gamma$ be the topological monoid of equivariant homotopy equivalences from the unit sphere $S(V)$ to itself. Write $G(V)^\Gamma/O(V)^\Gamma$ for the homotopy fiber of the map of classifying spaces, and write $G^\Gamma/O^\Gamma$ for the colimit of this over all $V$ in a complete universe (or the sequential colimit over direct sums of copies of the regular representation).

I think maybe with some work I could get a good connectivity statement for the map $G(V)^\Gamma/O(V)^\Gamma\to G^\Gamma/O^\Gamma$, but I wonder if someone has already done so, and more generally if anyone has any insights to offer about any of this.

Note that $BO^\Gamma$ is the product, over irreducible real representations $W$, of either $BO$, $BU$, or $BSp$ depending on the type of $W$, while $G^\Gamma$ is the union of the invertible components of $(\Omega^\infty S^\infty)^\Gamma$. In particular, $\pi_1(BG^\Gamma)$ is the group of units of the Burnside ring of $\Gamma$.