Let $G$ be any finite group. I'm trying to show that $G$ has a nilpotent subgroup $H$ such that $H^G:=\langle H^g : g \in G \rangle = G$, i.e. the normal closure of $H$ in $G$ is the whole group.
I know that $H$ cannot have a trivial center since it is nilpotent. Also $H$ cannot be normal for otherwise $H^g=H$ for all $g \in G$.
Other than that, I'm stuck for ideas. How do I construct such a subgroup $H$?