Free-abelian by free groups: homomorphisms and algorithmic explorations

We obtain an explicit description of the endomorphisms of free-abelian by free groups together with a characterization of when they are injective and surjective. As a consequence we see that free-abelian by free groups are Hopfian and not coHopfian, and we investigate the isomorphism problem and the Brinkmann Problem for this family of groups. In particular, we prove that the isomorphism problem (undecidable in general) is decidable when restricted to finite actions, and that the Brinkmann Problem is decidable both for monomorphisms and automorphisms.