  Let $p$ be a prime number, $G$ a finite group, $P$ a $p$-subgroup of $G$ and
$k$ an algebraically closed field of characteristic $p$. We study the
relationship between the category $\Ff_P(G)$ and the behavior of
$p$-permutation $kG$-modules with vertex $P$ under the Brauer construction. We
give a sufficient condition for $\Ff_P(G)$ to be a saturated fusion system. We
prove that for Scott modules with abelian vertex, our condition is also
necessary. In order to obtain our results, we prove a criterion for the
categories arising from the data of $(b, G)$-Brauer pairs in the sense of
Alperin-Brou\'e and Brou\'e-Puig to be saturated fusion systems on the
underlying $p$-group.
