# Restricted wreath product as fundamental group of a space with coinciding Reidemeister and Nielsen numbers

I am studying a group $$\mathbb{Z}_n \wr \mathbb{Z}^k$$, where $$\wr$$ denotes the restricted wreath product: $$\mathbb{Z}_n \wr \mathbb{Z}^k = \bigoplus_{x\in\mathbb{Z}^k}(\mathbb{Z_n})_x\rtimes\mathbb{Z^k},$$ where each $$(\mathbb{Z_n})_x=\mathbb{Z}_n$$ and $$\mathbb{Z}^k$$ acts with shifts.

I want to determine whether this group is a fundamental group of some Jiang-type topological space $$X$$, i. e. for every selfmap $$f\colon X \to X$$ either $$R(f)=\infty$$ or $$R(f)=N(f)$$, where $$R(f)$$ and $$N(f)$$ are the Reidemester and Nielsen numbers respectively. It is known that nilmanifolds and $$H$$-spaces are of Jiang-type.

I came across the Borel construction, which has unrestricted wreath product as its fundamental group, but I need the restricted one.