  Let $G,D_{0},D_{1}$ be finite groups such that $D_{0}\trianglelefteq D_{1}$
are groups of automorphisms of $G$ that contain the inner automorphisms of $G$.
Assume that $D_{1}/D_{0}$ has a normal $2$-complement and that $D_{1}$ acts
fixed-point-freely on the set of $D_{0}$-conjugacy classes of involutions of
$G$ (i.e., $C_{D_{1}}(a)D_{0}<D_{1}$ for every involution $a\in G$). We prove
that $G$ is solvable. We also construct a nonsolvable finite group that
possesses no characteristic conjugacy class of nontrivial cyclic subgroups.
This shows that an assumption on the structure of $D_{1}/D_{0}$ above must be
made in order to guarantee the solvability of $G$ and also yields a negative
answer to Problem 3.51 in the Kourovka Notebook, posed by A. I. Saksonov in
1969.
