The behavior of $\pi$-submaximal subgroups under homomorphisms with $\pi$-separable kernels

We explore the extent to which constructing the inductive theory of $\mathfrak{X}$-submaximal subgroups is possible. To this end, we study the behavior of $\mathfrak{X}$-submaximal subgroups under homomorphisms with $\mathfrak{X}$-separable kernels and construct examples where such behavior is irregular.