  This note shows that the $n$-th homology suspension is surjective for certain
quotients of finite $((Z/2)^n$-CW.complexes. This is true as soon as the
equivariant $((Z/2)^n$-cohomology are quotients is a free
$H^*$((Z/2)^n$-module. An application is given to certain Brown-Gitler
spectrum.
