Reidemeister classes in lamplighter type groups

We prove that for any automorphism $\varphi$ of the restricted wreath product ${\mathbb{Z}}_2\mathrm{w}\mathrm{r}{\mathbb{Z}}^k$ and $$\mathbb{Z}$_3 $\mathrm{w}\mathrm{r}$ $\mathbb{Z}$^{2d}$theReidemeisternumber$R(\phi)$ is infinite, i.e. these groups have the property $R_{\mathrm{\infty }}$. For ${\mathbb{Z}}_3\mathrm{w}\mathrm{r}{\mathbb{Z}}^{2d+1}$ and $$\mathbb{Z}$_p $\mathrm{w}\mathrm{r}$ $\mathbb{Z}$^k$,where$p>3$ is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property $R_{\mathrm{\infty }}$. For these groups and ${\mathbb{Z}}_m\mathrm{w}\mathrm{r}\mathbb{Z}$, where $m$ is relatively prime to $6$, we prove the twisted Burnside-Frobenius theorem (TBFT${}_f$): if $R(\varphi )<\mathrm{\infty }$, then it is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action $[\rho ]\mapsto [\rho \circ \varphi ]$.