$\omega\sigma$-fibered Fitting classes

The paper considers only finite groups. A class of groups $\mathfrak F$ is called a Fitting class if it is closed under normal subgroups and products of normal $\mathfrak F$-subgroups; formation, if it is closed with respect to factor groups and subdirect products; Fitting formation if $\mathfrak F$ is a formation and Fitting class at the same time.
For a nonempty subset $\omega$ of the set of primes $\mathbb P$ and the partition $\sigma =\{\sigma_i\mid i\in I\}$, where $\mathbb P=\cup_{i\in I}\sigma _i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\not =j$, we introduce the $\omega\sigma R$-function $f$ and $\omega\sigma FR$-function $\varphi$. The domain of these functions is the set $\omega\sigma\cup\{\omega'\}$, where $\omega\sigma=\{ \omega\cap\sigma_i\mid\omega\cap\sigma_i\not =\varnothing\}$, $\omega'=\mathbb P\setminus\omega$. The range of function values is the set of Fitting classes and the set of nonempty Fitting formations, respectively. The functions $f$ and $\varphi$ are used to determine the $\omega\sigma$-fibered Fitting class $\mathfrak F=\omega\sigma R(f,\varphi)=(G: O^{\omega} (G)\in f(\omega' )$ and $G^{\varphi (\omega\cap\sigma_i )} \in f(\omega\cap\sigma_i )$ for all $\omega\cap\sigma_i \in\omega\sigma (G))$ with the $\omega\sigma$-satellite $f$ and the $\omega\sigma$-direction $\varphi$.
The paper gives examples of $\omega\sigma$-fibered Fitting classes. Two types of $\omega\sigma$-fibered Fitting classes are distinguished: $\omega\sigma$-complete and $\omega\sigma$-local Fitting classes. Their directions are indicated by $\varphi_0$ and $\varphi_1$, respectively. It is shown that each nonempty nonidentity Fitting class is an $\omega\sigma$-complete Fitting class for some nonempty set $\omega\subseteq\mathbb P$ and any partition $\sigma$. A number of properties of $\omega\sigma$-fibered Fitting classes are obtained. In particular, a definition of an internal $\omega\sigma$-satellite is given and it is shown that each $\omega\sigma$-fibered Fitting class always has an internal $\omega\sigma$-satellite. For $\omega=\mathbb P$, the concept of a $\sigma$-fibered Fitting class is introduced. The connection between $\omega\sigma$-fibered and $\sigma$-fibered Fitting classes is shown.