Stabilization of the homotopy groups of the self equivalences of linear spheres

Let $G$ be a finite group. Let $U_1,U_2,dots$ be a sequence of orthogonal representations in which any irreducible representation of $oplus_{n geq 1} U_n$ has infinite multiplicity. Let $V_n=oplus_{i=1}^n U_n$ and $S(V_n)$ denote the linear sphere of unit vectors. Then for any $i geq 0$ the sequence of group $dots rightarrow pi_i operatorname{map}^G(S(V_n),S(V_n)) rightarrow pi_i operatorname{map}^G(S(V_{n+1}),S(V_{n+1})) rightarrow dots$ stabilizes with the stable group $oplus_H omega_i(BW_GH)$ where $H$ runs through representatives of the conjugacy classes of all the isotropy group of the points of $S(oplus_n U_n)$.