We explore the existence of homomorphisms between outer automorphism groups of free groups $\out\to\outm$. We prove that if $n>8$ is even and $n\neq m\le 2n$, or $n$ is odd and $n\neq m\le 2n-2$, then all such homomorphisms have finite image; in fact they factor through ${\rm{det}}\colon \out\to \Z/2$. In contrast, if $m=r^n(n-1)+1$ with $r$ coprime to $(n-1)$, then there exists an embedding $\out\hookrightarrow\outm$. In order to prove this last statement, we determine when the action of $\out$ by homotopy equivalences on a graph of genus $n$ can be lifted to an action on a normal covering with abelian Galois group.