17.36 (2010)
SolvedTwo groups are called isospectral if they have the same set of element orders. Find all finite non-abelian simple groups $G$ for which there is a finite group $H$ isospectral to $G$ and containing a proper normal subgroup isomorphic to $G$. For every simple group $G$ determine all groups $H$ satisfying this condition.
It is easy to show that a group $H$ must satisfy the condition $G < H \leqslant \text{Aut } G$.
Progress
All such groups are determined: for exceptional groups of Lie type in (M. A. Zvezdina, Algebra Logic, 55, no. 5 (2016), 354–366); for the other simple groups in (M. A. Grechkoseeva, Siberian Math. J., 59, no. 4 (2018), 623–640).
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