  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.
