$\Omega\zeta$-foliated Fitting classes

All groups under consideration are assumed to be finite. For a nonempty subclass of $\Omega$ of the class of all simple groups $\frak I$ and the partition $\zeta =\{\zeta_i\mid i\in I\}$, where $\zeta_i$ is a nonempty subclass of the class $\frak I$, $\frak I =\cup_ {i\in I}\zeta _i$ and $\zeta_i \cap \zeta_j = \varnothing$ for all $i\not = j$, $\Omega\zeta R$-function $f$ and $\Omega\zeta FR$-function $\varphi$ are introduced. The domain of these functions is the set $\Omega\zeta\cup \{\Omega '\}$, where $\Omega\zeta =\{\Omega\cap\zeta_i\mid\Omega\cap\zeta_i\not =\varnothing\}$, $\Omega '=\frak I\setminus\Omega$. The scope of these 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 $\Omega\zeta$-foliated Fitting class $\frak F=\Omega\zeta R(f,\varphi )=(G: O^\Omega (G)\in f(\Omega' )$ and $G^{\varphi (\Omega\cap\zeta_i )}\in f(\Omega\cap\zeta_i )$ for all $\Omega\cap\zeta_i \in\Omega\zeta (G))$ with $\Omega\zeta$-satellite $f$ and $\Omega\zeta$-direction $\varphi$. The paper gives examples of $\Omega\zeta$-foliated Fitting classes. Two types of $\Omega\zeta$-foliated Fitting classes are defined: $\Omega\zeta$-free and $\Omega\zeta$-canonical Fitting classes. Their directions are indicated by $\varphi_0$ and $\varphi_1$ respectively. It is shown that each non-empty non-identity Fitting class is a $\Omega\zeta$-free Fitting class for some non-empty class $\Omega\subseteq\frak I$ and any partition $\zeta$. A series of properties of $\Omega\zeta$-foliated Fitting classes is obtained. In particular, the definition of internal {$\Omega\zeta$-sa}tellite is given and it is shown that every $\Omega\zeta$-foliated Fitting class has an internal $\Omega\zeta$-satellite. For $\Omega=\frak I$, the concept of a $\zeta$-foliated Fitting class is introduced. The connection conditions between $\Omega\zeta$-foliated and $\zeta$-foliated Fitting classes are shown.