$G$-Covering systems of subgroups for classes of $p$-supersoluble and $p$-nilpotent finite 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 $G\in\mathscr{F}$ whenever either $\Sigma=\varnothing$ or $\Sigma\ne\varnothing$ and every subgroup in $\Sigma$ belongs to $\mathscr{F}$. We find the systems of subgroups in the class of finite soluble groups $G$ which are simultaneously the $G$-covering systems of subgroups for the classes of $p$-supersoluble and $p$-nilpotent groups.