1.73 (1965)

Not always. The extension $H$ of the free group on free generators $a, b$ by the automorphism $\varphi : a \to a^{-1}$, $b \to b$ is a linear group over the integers. For every $n \in \mathbb{Z}$ the element $c_n = \varphi b^{-n} a b^n$ has order 2 and $C_H(c_n) = \langle c_n \rangle$. Let the dash denote an isomorphism of $H$ onto its copy $H'$. As a counterexample one can take the subgroup $G \leqslant H \times H'$ generated by the elements $a, \varphi, a'\varphi'$, $b b'$. Indeed, $G$ contains the subgroups $T_n = \langle c_0, c'_n \rangle, n \in \mathbb{Z}$, which are maximal periodic (even in $H \times H'$). If $T_n$ and $T_m$ are conjugate by an element $x y' \in G$ (where $x \in H$ and $y' \in H'$), then $c_0^x = c_0$ and $c'_n = c_m^{y'}$, whence $x \in \langle c_0 \rangle$, $y' \in b^{m-n} \langle c_m \rangle$. Since $xy' \in G$, the sums of the exponents at the occurrences of $b$ in $x$ and $y$ (in any expression) must coincide; hence $m = n$. (Yu. I. Merzlyakov, 1973.)