2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/135041Let $\Out(F_n)$ denote the outer automorphism group of the free group $F_n$ with $n>3$. We prove that for any finite index subgroup $Γ<\Out(F_n)$, the group $\Aut(Γ)$ is isomorphic to the normalizer of $Γ$ in $\Out(F_n)$. We prove that $Γ$ is {\em co-Hopfian} : every injective homomorphism $Γ\to Γ$ is surjective. Finally, we prove that the abstract commensurator $\Comm(\Out(F_n))$ is isomorphic to $\Out(F_n)$.Revised version, 43 pages. To appear in Publ. Math. IHESGroup Theory20F28text