When is the search of relatively maximal subgroups reduced to quotients?

Let ${\mathfrak{X}}$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Denote by ${\mathrm{k}}_{\mathfrak{X}}(G)$ the number of conjugacy classes ${\mathfrak{X}}$-maximal subgroups of a finite group $G$. The natural problem to describe up to conjugacy ${\mathfrak{X}}$-maximal subgroups of a given finite group is complicated by the fact that it is not inductive. In particular, generally speaking, the image of an ${\mathfrak{X}}$-maximal subgroup is not ${\mathfrak{X}}$-maximal in the image of a homomorphism. Nevertheless, there are group homomorphisms which preserve the number of conjugacy classes of ${\mathfrak{X}}$-maximal 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 classes of ${\mathfrak{X}}$-maximal subgroups of the image and preimage. All such homomorphisms are completely described in the paper. More precisely, it is proved that, for a homomorphism $\phi$ from a group $G$, the equality ${\mathrm{k}}_{\mathfrak{X}}(G)={\mathrm{k}}_{\mathfrak{X}}(\mathrm{im}\, \phi)$ 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 $\phi$ belong to an explicitly given list.