  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.
  If a general automorphism $\varphi$ of $G\wr \mathbb Z^k$ has a finite order
(this is the case for $\varphi$ detected in the first part of the paper) and
$R(\varphi)<\infty$, we prove that $R(\varphi)$ coincides with the number of
equivalence classes of finite-dimensional irreducible unitary representations
of $G\wr \mathbb Z^k$, which are fixed by the dual map $[\rho]\mapsto
[\rho\circ \varphi]$ (i.e. we prove the conjecture about finite twisted
Burnside-Frobenius theorem, TBFT$_f$, for these $\varphi$).
