$G$-Covering Subgroup Systems for the Class of All $\sigma$-Nilpotent Finite Groups

Let $\mathfrak F$ be a nonempty class of groups and let $G$ be a finite group. A set $\Sigma$ of subgroups of the group $G$ is called a $G$-covering subgroup system for the class $\mathfrak F$ (or an $\mathfrak F$-covering subgroup system of $G$) if $\Sigma \subseteq \mathfrak F$ always implies that $G \in \mathfrak F$. In this paper, a nontrivial set of subgroups of $G$ is constructed which is a $G$-covering subgroup system for the class $\mathfrak F$ of all $\sigma$-nilpotent groups.