$\sigma_{\Omega}$-foliated Fitting classes of multioperator $T$-groups and its satellites

In 2001, V.A. Vedernikov and M.M. Sorokina proposed a functional approach to the construction of formations and fitting classes of finite groups by considering, in addition to satellite functions, another type of functions - directions. As a result, $\omega$-fibered ($\Omega$-foliated) formations and Fitting classes of finite groups were constructed, including the well-known $\omega$-local ($\Omega$-composite) formations and Fitting classes, where $\omega$ is a nonempty set of primes ($\Omega$ — a nonempty subclass of the class of all simple groups). Further research has shown that the concept of foliation can be applied to the construction of foliated formations and Fitting classes of multioperator $T$-groups with finite composition series. A new idea in the functional approach of constructing classes of groups was proposed by A.N. Skiba. In a series of articles, he developed the $\sigma$-theory of finite groups, where $\sigma$ is an arbitrary partition of the set of all primes, and applied its methods to the construction of $\sigma$-local formations. Classes generalizing $\omega$-fibered and $\Omega$-foliated formations and Fitting classes of finite groups were constructed on the basis of the $\sigma$-methods. We define classes generalizing the foliated Fitting classes of multioperator T-groups with composite series and study their minimal and inner satellites.