  A finitely generated group $G$ is called strongly scale-invariant if there
exists an injective homomorphism $f:G\to G$ such that $f(G)$ is a finite index
subgroup of $G$ and such that $\cap_{n\geq 0} f^n(G)$ is finite. Nekrashevych
and Pete conjectured that all strongly scale-invariant groups are virtually
nilpotent, after disproving a stronger conjecture by Benjamini.
  This conjecture is known to be true in some situations. Der\'e proved it for
virtually polycyclic groups. In this paper, we provide an elementary proof for
those polycyclic groups that can be written as a semi-direct product
$\mathbb{Z}^n\rtimes \mathbb{Z}$.
