On Shemetkov's Question about the $\mathfrak{F}$-Hypercenter

The principal factor $H/K$ of a group $G$ is said to be $\mathfrak{F}$-central if
$$ (H/K)\rtimes (G/C_G(H/K))\in\mathfrak{F}. $$
The $\mathfrak{F}$-hypercenter of a group $G$ is defined to be a maximal normal subgroup of $G$ such that all $G$-principal factors below it are $\mathfrak{F}$-central in $G$. In 1995, at the Gomel algebraic seminar, L. A. Shemetkov formulated the problem of describing formations of finite groups $\mathfrak{F}$ for which, in any group, the intersection of $\mathfrak{F}$-maximal subgroups coincides with the $\mathfrak{F}$-hypercenter. In this paper, new properties of such formations are obtained. In particular, a series of hereditary unsaturated formations of soluble groups is constructed, which are the answers to Shemetkov's problem.