One application of the $\sigma$-local formations of finite groups

Throughout this paper, all groups are finite. Let $\sigma ={\sigma_{i} | i\in I }$besomepartitionofthesetofallprimes$\Bbb{P}$\mathrm{.}If$n$isaninteger,thesymbol$\sigma (n)$ denotes the set ${\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset }$\mathrm{.}Theintegers$n$and$m$arecalled$\sigma$-coprimeif$\sigma (n)\cap \sigma (m)=\emptyset$. Let $t>1$ be a natural number and let $\mathfrak{F}$ be a class of groups. Then we say that $\mathfrak{F}$ is ${\mathrm{\Sigma }}_t^{\sigma }$-closed provided $\mathfrak{F}$ contains each group $G$ with subgroups $A_{1}, \ldots , A_{t}\in \mathfrak{F}$whoseindices$|G:A_{1}|$,$\ldots$,$|G:A_{t}|$arepairwise$\sigma$-coprime. In this paper, we study ${\mathrm{\Sigma }}_t^{\sigma }$-closed classes of finite groups.