Let \Out(F_n) denote the outer automorphism group of the free group Fn
with n>3. We prove that for any finite index subgroup \Gamma<\Out(F_n), the
group \Aut(\Gamma) is isomorphic to the normalizer of Γ in
\Out(F_n). We prove that Γ is {\em co-Hopfian} : every injective
homomorphism Γ→Γ is surjective. Finally, we prove that the
abstract commensurator \Comm(\Out(F_n)) is isomorphic to \Out(F_n).Comment: Revised version, 43 pages. To appear in Publ. Math. IHE