Crossed modules as maps between connected components of topological groups

The purpose of this paper is to observe that a homomorphism of discrete groups f : Γ → G arises as the induced map π 0 (M) → π 0 (M) on path components of some closed normal inclusion of topological groups M ⊂ M if and only if the map f can be equipped with a crossed module structure. In that case an essentially unique realization M _M exists by homotopically discrete topological groups. The results here are topological elaboration and exposition using existing simplicial techniques.