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.