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.