On Questions Posed by Shemetkov, Ballester-Bolinches, and Perez-Ramos in Finite Group Theory

A chief factor $H/K$ of a group $G$ is said to be $\mathfrak{F}$-central if $(H/K)\rtimes (G/C_G(H/K))\in\mathfrak{F}$. In 1997, Shemetkov posed the problem of describing finite group formations $\mathfrak{F}$ such that $\mathfrak{F}$ coincides with the class of groups for which all chief factors are $\mathfrak{F}$-central. We refer to such formations as centrally saturated. We prove that the centrally saturated formations form a complete distributive lattice. As an answer to a question posed by Ballester-Bolinches and Perez-Ramos, conditions for a centrally saturated formation to be saturated and solvably saturated in the class of all groups are found. As a consequence, a criterion for hereditary Fitting formations to be solvably saturated is obtained.