  Reidemeister (or twisted conjugacy) classes are considered in restricted
wreath products of the form $G\wr \mathbb{Z}^k$, where $G$ is a finite group.
  For an automorphism $\varphi$ of finite order (supposed to be the same for
the torsion subgroup $\oplus G$ and the quotient $\mathbb{Z}^k$) with finite
number $R(\varphi)$ of Reidemeister classes, this number is identified with the
number of equivalence classes of finite-dimensional unitary irreducible
representations of the product that are fixed by the dual homeomorphism
$\widehat{\varphi}$ (i.e. the so-called conjecture TBFT$_f$ is proved in this
case).
  For these groups and automorphisms, we prove the following conjecture: if a
finitely generated residually finite group has an automorphism with
$R(\varphi)<\infty$ then it is solvable-by-finite (so-called conjecture R).
