Abelian covers of graphs and maps between outer automorphism groups of free groups

We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F (n) ) -> Out(F (m) ). We prove that if n > 8 is even and n not equal m a parts per thousand currency sign 2n, or n is odd and n not equal m a parts per thousand currency sign 2n - 2, then all such homomorphisms have finite image; in fact they factor through det : . In contrast, if m = r (n) (n - 1) + 1 with r coprime to (n - 1), then there exists an embedding . 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.