Generalized rank composition formations of finite groups

In this paper one construction of composition formations was introduced. This construction contains formations of quasi-$\mathfrak{F}$-groups, $c$-supersoluble groups and groups defined by ranks of chief factors. The structure of groups from introduced formations was described. As corollaries some results of different authors were obtained. A question of L.A. Shemetkov about the intersection of $\mathfrak{F}$-maximal subgroups and the $\mathfrak{F}$-hypercenter was investigated for these formations.