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.
Categories: math.GR