The reduction theorem for relatively maximal subgroups

Let (sic) 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 C then the image of an (sic)-maximal subgroup H of G in G/A is not, in general, (sic)-maximal in G/A. We say that the reduction (sic)-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 (sic)-maximal subgroups in G and G/A is the same. The reduction (sic)-theorem for A implies that HA/A is (sic)-maximal in G/A for every extension G of A and every (sic)-maximal subgroup H of G. In this paper, we prove that the reduction (sic)-theorem holds for A if and only if all (sic)-maximal subgroups of A are conjugate in A and classify the finite groups with this property in terms of composition factors.