Endomorphisms of Automorphism Groups of Free Groups

It is proved that any non-trivial endomorphism of an automorphism group $\operatorname{Aut}F_n$ of a free group $F_n$, for $n\geqslant3$, either is an automorphism or factorization over a proper automorphism subgroup. An endomorphism of $\operatorname{Aut}F_2$ is an automorphism, or else a homomorphism onto one of the groups $S_3$, $D_8$, $Z_2\times Z_2$, $Z_2$, or $S_3*_{Z_2}(Z_2\times Z_2)$. A non-trivial homomorphism of $\operatorname{Aut}F_n$ into $\operatorname{Aut}F_m$, for $n\geqslant3$, $m\geqslant2$, and $n>m$, is a homomorphism onto $Z_2$ with kernel $\operatorname{SAut}F_n$. As a consequence, we obtain that $\operatorname{Aut}F_n$ is co-Hopfian.