$\bar \omega$-fibered formations of finite groups

Only finite groups are considered. The work is devoted to the study of formations which are classes of groups that are closed with respect to homomorphic images and subdirect products. For a non-empty set $\omega$ of primes V.A. Vedernikov, using two types of functions, defined $\omega$-fibered formations of finite groups. Developing this functional approach, in the paper for an arbitrary partition $\bar\omega$ of the set $\omega$ we constructed $\bar\omega$-fibered formations. The construction uses the $\sigma$-concept of A.N. Skiba for the study of finite groups and their classes, where $\sigma$ is an arbitrary partition of the set $\mathbb P$ of all primes. We gave examples of $\bar\omega$-fibered formations, established their properties (existence of $\bar\omega$-satellites of different types; sufficient conditions for a group $G$ to belong to an $\bar\omega$-fibered formation; relationship with $\omega$-fibered and $\mathbb P_{\sigma}$-fibered formations).