On the $\pi$-potency of descending HNN-extensions of groups

Let $G$ be a group, let $\varphi$ be an isomorphism of $G$ onto a subgroup $K$ of $G$, and let $G^*$ be a descending HNN-extension of $G$ corresponding to $\varphi $. The potency of $G$ is not inherited by $G^*$ even in the simplest case, when $G$ is an infinite cyclic group. We prove that if $G$ is a finitely generated torsion-free nilpotent group (a polycyclic group); then the index $m = [G : K]$ of $K$ in $G$ is finite and $G^*$ is $\pi $-potent (virtually $\pi $-potent), where $\pi $ is the set of all primes greater than $m$. We also prove some generalizations of this assertion. Some of the results of this work on the potency of descending HNN-extensions are analogs of the well-known theorems on the residual finiteness of the HNN-extensions.