A covering subgroup system for the class of $p$-nilpotent groups

Let $\mathscr F$ be a class of groups and let $G$ be a finite group. We call a set $\Sigma$ of subgroups of $G$ a covering subgroup system of $G$ for $\mathscr F$ (or directly an $\mathscr F$-covering subgroup system of $G$) if $G\in\mathscr F$ whenever every subgroup in $\Sigma$ is in $\mathscr F$. We give some covering subgroup systems for the class of all $p$-nilpotent groups.