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