Multiply $\omega$-Local Formations and Fitting Classes of Finite Groups

The present article deals only with finite groups. Functions of the form
$$ f\colon\omega\cup\{\omega'\}\to\{\text{group formations}\} $$
are called $\omega$-local satellites (here $\omega$ denotes a nonempty set of primes). Functions of this form are used to study the structure of the $\omega$-local formations, i.e. the formations $\mathfrak{F}$ such that $G\in\mathfrak{F}$ whenever $G/\Phi(G)\cap O_{\omega}(G)\in\mathfrak{F}$. The theory of $\omega$-local Fitting classes is analyzed which is dual to the theory of $\omega$-local formations.