The reduction theorem for relatively maximal subgroups

Let $\mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $\mathfrak{X}$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $\mathfrak{X}$-maximal in $G/A$. We say that the reduction $\mathfrak{X}$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a normal subgroup), the number of conjugacy classes of $\mathfrak{X}$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $\mathfrak{X}$-theorem for $A$ implies that $HA/A$ is $\mathfrak{X}$-maximal in $G/A$ for every extension $G$ of $A$ and every $\mathfrak{X}$-maximal subgroup $H$ of $G$. In this paper, we prove that the reduction $\mathfrak{X}$-theorem holds for $A$ if and only if all $\mathfrak{X}$-maximal subgroups are conjugate in $A$ and classify the finite groups with this property in terms of composition factors.

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