  In this paper one construction of composition formations was introduced. This
construction contains formations of quasinilpotent groups, $c$-supersoluble
groups, groups defined by ranks of chief factors and some new classes of
groups. A partial answer on a question of L.\,A. Shemetkov about the
intersection of $\mathfrak{F}$-maximal subgroups and the
$\mathfrak{F}$-hypercenter was given for these composition formations.
