Formations of finite groups in polynomial time ii: the $\mathfrak{F}$-hypercenter and its generalizations

For a wide family of formations $\mathfrak{F}$ (which includes Baer-local formations) of finite groups it is proved that the $ \mathfrak{F}$-hypercenter of a permutation finite group of degree $n$ can be computed in polynomial time in $n$. In particular, the algorithms for computing the $\mathfrak{F}$-hypercenter for the following classes of groups are suggested: hereditary local formations with the Shemetkov property, rank formations, formations of all quasinilpotent, Sylow tower of type $\varphi$, $p$-nilpotent, supersoluble, $w$-supersoluble and $SC$-groups. For some of these formations $\mathfrak{F}$ algorithms for the computation of the intersection of all maximal $\mathfrak{F}$-subgroups of a finite group are suggested.