Examples of Nonpronormal Relatively Maximal Subgroups of Finite Simple Groups

Using R. Wilson’s recent results, we prove the existence of triples \((\mathfrak{X},G,H)\) such that \(\mathfrak{X}\) is a complete (i.e., closed under taking subgroups, homomorphic images, and extensions) class of finite groups, \(G\) is a finite simple group, and \(H\) is its \(\mathfrak{X}\)-maximal subgroup nonpronormal in \(G\). This disproves a conjecture stated earlier by the second author and W. Guo.