We say a group $ G $ is a Wang group if it fits into a SES $$ 0 \to N \to G \to \mathbb{Z}^k \to 0 $$ where $ N $ is nilpotent finitely generated and torsion free. Such a group $ G $ is always solvable finitely generated and torsion free.
Is it true that the commutator subgroup of a Wang group $ G $ is finite if and only if $ G $ is abelian?
when $ N $ is abelian the answer seems to be yes see the comment from markvs here
Semidirect product of free abelian groups
I am interested in the more general case where $ N $ is nilpotent but perhaps not abelian.