Reidemeister classes in wreath products of abelian groups

Among restricted wreath products $G\wr \mathbb{Z}^k $, where $G$ is a finite abelian group, we find three large classes of groups admitting an automorphism $\varphi$ with finite Reidemeister number $R(\varphi)$ (number of $\varphi$-twisted conjugacy classes). In other words, groups from these classes do not have the $R_\infty$ property.
Moreover, we prove that if $\varphi$ is a finite order automorphism of $G\wr \mathbb{Z}^k$ with $R(\varphi)<\infty$, then $R(\varphi)$ is equal to the number of fixed points of the map $[\rho]\mapsto [\rho\circ \varphi]$ defined on the set of equivalence classes of finite dimensional irreducible unitary representations of $G\wr \mathbb{Z}^k$.