Finite nilpotent group implies every fusion system-equivalent group is isomorphic

Statement
Suppose $$G_1$$ and $$G_2$$ are fusion system-equivalent finite groups. Suppose further that $$G_1$$ is a finite nilpotent group. Then, $$G_1$$ and $$G_2$$ are isomorphic groups. In particular, $$G_2$$ is also a finite nilpotent group.

Proof
The idea is to note that a finite group is nilpotent if and only if the fusion system induced on every Sylow subgroup is the inner fusion system for that subgroup.