Characteristic classes of involutions in nonsolvable groups

Let $G,D_0,D_1$ be finite groups such that $D_0⊴D_1$ are groups of automorphisms of $G$ that contain the inner automorphisms of $G$. Assume that $D_1\mathrm{/}{D}_0$ 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 $D_1\mathrm{/}{D}_0$ 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.