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

Only finite groups are considered. The notion of $\omega$-local formation of groups, where $\omega$ is a nonempty set of primes, is a well-known generalization of the notion of local formation. For an arbitrary partition $\sigma$ of the set of all primes, A. N. Skiba developed the $\sigma$-theory of finite groups and applied its methods for constructing $\sigma$-local formations. The concept of $\omega$-fiberedness introduced by V. A. Vedernikov for classes of groups made it possible to construct an infinite series of $\omega$-fibered formations, while $\omega$-local formations formed one of the types of this series. In this paper, we study $\bar\omega$‑fibered formations of groups, where $\bar\omega$ is an arbitrary partition of the set $\omega$, constructed on the basis of Skiba's $\sigma$-approach applied to $\omega$-fibered formations. Consider functions $f\colon{\bar{\omega}} \cup \{\bar{\omega}'\}\rightarrow \{$formations of groups$\}$ and $\gamma\colon\bar{\omega} \rightarrow \{$nonempty Fitting formations of groups$\}$, where $f(\bar{\omega}')\not=\varnothing$ and the class of groups $\gamma(\omega_{i})$ contains all ${\omega_{i}}'$-groups for any $\omega_{i} \in \bar{\omega}$. A formation $\frak F = (G \in \frak G \vert G/O_{\omega}(G) \in f(\bar{\omega}')$ and $G/G_{\gamma (\omega_{i})} \in f (\omega_{i})$ for any $\omega_{i} \in \bar{\omega} \cap \pi (G))$ is called an $\bar{\omega}$-fibered formation with direction $\gamma$ and $\bar{\omega}$-satellite $f$. In this paper we study inner $\bar\omega$-satellites of $\bar\omega$-fibered formations, i.e., $\bar\omega$-satellites whose values are contained in the considered formation. The following problems are solved: the existence of a canonical $\bar\omega$-satellite of an $\bar\omega$-fibered formation is proved, and the structure of a maximal inner $\bar\omega$-satellite of an $\bar\omega$-fibered formation is described.