Criteria for supersolvability of saturated fusion systems

Let $p$ be a prime number. A saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is said to be supersolvable if there is a series $1 = S_0 \le S_1 \le \dots \le S_m = S$ of subgroups of $S$ such that $S_i$ is strongly $\mathcal{F}$-closed for all $0 \le i \le m$ and such that $S_{i+1}/S_i$ is cyclic for all $0 \le i < m$. We prove some criteria that ensure that a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is supersolvable provided that certain subgroups of $S$ are abelian and weakly $\mathcal{F}$-closed. Our results can be regarded as generalizations of purely group-theoretic results of Asaad.