An example of a relatively maximal nonpronormal subgroup of odd order in a finite simple group

We prove the existence of a triple $({\mathfrak X},G,H)$, where ${\mathfrak X}$ is a class of finite groups consisting of groups of odd order which is complete (i.e., closed under subgroups, homomorphic images, and extensions), $G$ is a finite simple group, $H$ is an ${\mathfrak X}$-maximal subgroup in $G$, and $H$ is not pronormal in $G$.