On the $K$-theory of groups of the form ${\mathbb{Z}}^n⋊\mathbb{Z}\mathrm{/}m$ with $m$ square-free

We provide an explicit computation of the topological $K$-theory groups $K_{\ast }(C_r^{\ast }({\mathbb{Z}}^n⋊\mathbb{Z}\mathrm{/}m))$ of semidirect products of the form ${\mathbb{Z}}^n⋊\mathbb{Z}Z\mathrm{/}m$ with $m$ square-free. We want to highlight the fact that we are not impossing any conditions on the $\mathbb{Z}\mathrm{/}m$-action on ${\mathbb{Z}}^n$. This generalizes previous computations of Lück-Davis and Langer-Lück.