$G$-covering systems of subgroups for the class of supersoluble groups

Let $\mathscr{F}$ be a class of groups. Given a group $G$, assign to $G$ some set of its subgroups $\Sigma=\Sigma(G)$. We say that $\Sigma$ is a $G$-covering system of subgroups for $\mathscr{F}$ (or, in other words, an $\mathscr{F}$-covering system of subgroups in $G$) if $G\in\mathscr{F}$ wherever either $\Sigma=\varnothing$ or $\Sigma\ne\varnothing$ and every subgroup in $\Sigma$ belongs to $\mathscr{F}$. In this paper, we provide some nontrivial sets of subgroups of a finite group $G$ which are $G$-covering subgroup systems for the class of supersoluble groups. These are the generalizations of some recent results, such as in [1–3].