  A subgroup $A$ of a finite group $G$ is said to be a $CAP$-subgroup of $G$,
if for any chief factor $H/K$ of $G$, either $A H= AK$ or $A\cap H = A \cap K$.
Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion
system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there
exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$,
such that $S_{i+1}/S_i$ is cyclic, and $S_i$ is strongly $\mathcal{F}$-closed
for any $i=0,1,\cdots,n$. In this paper, we first introduce the concept of
strong $p$-$CAP$-subgroups, and investigate the properties of finite groups
under the assumption that some subgroups of $G$ are partial $CAP$-subgroups or
strong $(p)$-$CAP$-subgroups, and obtain some criteria for a group $G$ to be
$p$-supersoluble. Then, we investigate the characterizations for
supersolvability of $\mathcal{F}_S (G)$ under the assumption that some
subgroups of $G$ are partial $CAP$-subgroups or strong $(p)$-$CAP$-subgroups of
$G$, and obtain some criteria for a fusion system $\mathcal{F}_S (G)$ to be
$p$-supersolvable.
