21.111 (2026)

Let $S$ be a finite simple nonabelian group that is not isomorphic to any group ${}^2B_2(q)$. A nonidentity automorphism $x$ of $S$ is called a $\tau$-automorphism if every two conjugates of $x$ in $\langle x, \text{Inn}(S) \rangle$ generate a subgroup of order not divisible by 3. If $S$ admits a $\tau$-automorphism, we call $S$ a $\tau$-group.
$\qquad$ a) List all $\tau$-groups up to isomorphism.
$\qquad$ b) Do $\tau$-automorphisms of odd order exist?