On the homotopy groups of the self equivalences of linear spheres

Let $S(V)$ be a complex linear sphere of a finite group $G$. %the space of unit vectors in a complex representation $V$ of a finite group $G$. Let $S(V)^{*n}$ denote the $n$-fold join of $S(V)$ with itself and let $aut_G(S(V)^*)$ denote the space of $G$-equivariant self homotopy equivalences of $S(V)^{*n}$. We show that for any $k geq 1$ there exists $M>0$ which depends only on $V$ such that $|pi_k aut_G(S(V)^{*n})| leq M$ is for all $n gg 0$.