Let $G$ be a finitely generated group and $N$ be a nilpotent group (not necessarily finitely generated) such that $N$ is a subgroup of $G$ and $G / N$ is $\mathbb{Z}$. That is, there is a short exact sequence $$ 1\rightarrow N \rightarrow G \rightarrow \mathbb{Z} \rightarrow 1$$ such that $G$ is finitely generated and $N$ is a nilpotent subgroup.
Is $G$ then supersolvable or at least polycyclic?
If $N$ were a finitely generated group, it is clearly supersolvable. But is it true even though $N$ is infinitely generated ensured that $G$ is finitely generated?
Any subgroup of a polycyclic group is finitely generated. By contraposition, if $N$ is not finitely generated, then $G$ is not polycyclic.