  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 $S(\oplus_n U_n)$.
