Fusion system-equivalence preserves perfectness

Statement

Suppose $G_1$ and $G_2$ are fusion system-equivalent finite groups. Suppose further that $G_1$ is a perfect group. Then, $G_2$ is also a perfect group.

Facts used

1. Focal subgroup theorem

Proof

The key idea is to show that a finite group is perfect if and only if, for every prime, the focal subgroup for that prime (i.e., the focal subgroup of its Sylow subgroup) is the whole Sylow subgroup. This follows from Fact (1).