Factorization of group homomorphisms

Let π be a torsion-free virtually polycyclic group. For any finitely generated group G, we show that there exists a torsion-free virtually polycyclic group image and an epimorphism image such that for any homomorphism phi : G→π, it factors through image, i.e., there exists a homomorphism image such that image. As an application, we give necessary and sufficient conditions for N(f,g)=R(f,g) to hold for maps f,g : X→Y between closed orientable n-manifolds where Y is an infrasolvmanifold, N(f,g) and R(f,g) denote the Nielsen and Reidemeister coincidence numbers, respectively.