On one criterion for the $\sigma$-nilpotency of a finite group

Let $\mathfrak F$ be a class of groups and let $G$ be a finite group. We refer to a set $\Sigma$ of subgroups of $G$ as a $G$-covering subgroup system for a class $\mathfrak F$ if $G \in \mathfrak F$ whenever $\Sigma \subseteq \mathfrak F$. Also, we provide some nontrivial $G$-covering subgroup system for the class $\mathfrak F$ of all $\sigma$-nilpotent groups.