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(Fn) → Out(Fm). We prove that if n > 8 is even and n ≠ m ≤ 2n, or n is odd and n ≠ m ≤ 2n − 2, then all such homomorphisms have finite image; in fact they factor through det : \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Out}(F_n) \to \mathbb{Z}/2}$$\end{document} . In contrast, if m = rn(n − 1) + 1 with r coprime to (n − 1), then there exists an embedding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Out}(F_n) \hookrightarrow {\rm Out}(F_m)}$$\end{document} . In order to prove this last statement, we determine when the action of Out(Fn) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group.