  Suppose, $G$ is a residually finite group of finite upper rank admitting an
automorphism $\varphi$ with finite Reidemeister number $R(\varphi)$ (the number
of $\varphi$-twisted conjugacy classes). We prove that such $G$ is
soluble-by-finite (in other words, any residually finite group of finite upper
rank, which is not soluble-by-finite, has the $R_\infty$ property).
  This reduction is the first step in the proof of the second main theorem of
the paper: suppose, $G$ is a residually finite group of finite Pr\"ufer rank
and $\varphi$ is its automorphism with $R(\varphi)<\infty$; then $R(\varphi)$
is equal to the number of equivalence classes of finite-dimensional irreducible
unitary representations of $G$, which are fixed points of the dual map
$\widehat{\varphi}:[\rho]\mapsto [\rho\circ \varphi]$ (i.e., we prove the
TBFT$_f$, the finite version of the conjecture about the twisted
Burnside-Frobenius theorem, for such groups).
