20.31 (2022)

Let $\omega(G)$ denote the set of element orders of a finite group $G$, and $h(G)$ the number of pairwise nonisomorphic finite groups $H$ with $\omega(H) = \omega(G)$. Find $h(L)$, where
$\qquad$ a) $L$ is the symmetric group of degree 10;
$\qquad$ b) $L$ is the automorphism group of the simple sporadic Janko group $J_2$;
$\qquad$ c) $\text{PSL}(2, q) < L \leqslant \text{Aut}(\text{PSL}(2, q))$.

See the current status in (M. A. Grechkoseeva, V. D. Mazurov, W. J. Shi, A. V. Vasil’ev, N. Yang, Commun. Math. Stat., 11, no. 2 (2023), 169–194; Subsection 4.2).