Maximal and submaximal $\mathfrak X$-subgroups

Let $\mathfrak X$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup $H$ of a finite group $G$ a submaximal $\mathfrak X$-subgroup if there exists an isomorphic embedding $\phi\colon G\hookrightarrow G^*$ of $G$ into some finite group $G^*$ under which $G^\phi$ is subnormal in $G^*$ and $H^\phi=K\cap G^\phi$ for some maximal $\mathfrak X$-subgroup $K$ of $G^*$. In the case where $\mathfrak X$ coincides with the class of all $\pi$-groups for some set $\pi$ of prime numbers, submaximal $\mathfrak X$-subgroups are called submaximal $\pi$-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal $\pi$-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal $\mathfrak X$- and $\pi$-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal $\mathfrak X$-subgroups are conjugate in a finite group $G$ in which all maximal $\mathfrak X$-subgroups are conjugate?