When is the search of relatively maximal subgroups reduced to quotient groups?

Let $\mathfrak{X}$ be a class finite groups closed under taking subgroups, homomorphic images, and extensions, andlet $\mathrm{k}_{\mathfrak{X}}(G)$ be the number of conjugacy classes $\mathfrak{X}$-maximal subgroups of a finite group $G$.The natural problem calling for a description, up to conjugacy, ofthe $\mathfrak{X}$-maximal subgroups of a given finite group is not inductive.In particular, generally speaking, the image of an $\mathfrak{X}$-maximalsubgroup is not $\mathfrak{X}$-maximal in the image of a homomorphism.Nevertheless, there exist group homomorphisms that preserve the number of conjugacy classes of maximal$\mathfrak{X}$-subgroups (for example, the homomorphisms whose kernels are $\mathfrak{X}$-groups).Under such homomorphisms, the image of an $\mathfrak{X}$-maximal subgroup is always $\mathfrak{X}$-maximal,and, moreover, there is a natural bijection between the conjugacy classesof $\mathfrak{X}$-maximal subgroups of the image and preimage.In the present paper, all such homomorphisms arecompletely described.More precisely, it is shown that, for a homomorphism $\phi$from a group $G$, the equality $\mathrm{k}_{\mathfrak{X}}(G)=\mathrm{k}_{\mathfrak{X}}(\operatorname{im} \phi)$holds if and only if $\mathrm{k}_{\mathfrak{X}}(\ker \phi)=1$,which in turn is equivalent to the fact that the composition factors of the kernel of $\phi$ lie in an explicitly given list.

finite group, complete class, -maximal subgroup, Hall subgroup, reduction -theorem