Corollary 1

*Suppose that $f$ is surjective. Let $\mathscr{G}$ (resp. $\mathscr{G}'$) be the set of stable (resp. normal stable) subgroups of $G$ containing $N$ and $\mathfrak{H}$ (resp. $\mathfrak{H}'$) the set of stable (resp. normal stable) subgroups of $H$, these sets being ordered by inclusion. The mapping $G' \mapsto f(G')$ is an ordered set isomorphism $\Phi : \mathscr{G} \to \mathfrak{H}$; the inverse isomorphism $\Psi : \mathfrak{H} \to \mathscr{G}$ is the mapping $H' \mapsto f^{-1}(H')$. Further $\Phi$ and $\Psi$ induce isomorphisms $\Phi' : \mathscr{G}' \to \mathfrak{H}'$ and $\Psi' : \mathfrak{H}' \to \mathscr{G}'$.