On the lattice of hereditary Shemetkov formations of finite groups in a class $\mathfrak{X}$

A formation $\mathfrak{F}$ of finite groups is called a Shemetkov formation in a class $\mathfrak{X}$ if every $\mathfrak{X}$-group not belonging to $\mathfrak{F}$, all of whose proper subgroups belong to $\mathfrak{F}$, is either a Schmidt group or a group of prime order. In this paper, for a hereditary solvably saturated formation $\mathfrak{X}$, it is proved that the lattice of all hereditary Shemetkov formations of $\mathfrak{X}$-groups in the class $\mathfrak{X}$ is lattice-isomorphic to the lattice of all subgraphs of some directed graph. As a corollary, a description of the lattice of all hereditary local Shemetkov formations of solvable groups in the class of all solvable groups is obtained, which was found by Ballester-Bolinches, Kamornikov, and Yi in 2024.