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.