  We prove that the restricted wreath product ${\mathbb{Z}_n
\mathbin{\mathrm{wr}} \mathbb{Z}^k}$ has the $R_\infty$-property, i. e. every
its automorphism $\varphi$ has infinite Reidemeister number $R(\varphi)$, in
exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$
divisible by 3. In the remaining cases there are automorphisms with finite
Reidemeister number, for which we prove the finite-dimensional twisted
Burnside--Frobenius theorem (TBFT): $R(\varphi)$ is equal to the number of
equivalence classes of finite-dimensional irreducible unitary representations
fixed by the action ${[\rho]\mapsto[\rho\circ\varphi]}$.
