Fusion system-equivalent finite groups

Definition
Suppose $$G_1$$ and $$G_2$$ are finite groups. We say that $$G_1$$ and $$G_2$$ are fusion system-equivalent if, for every prime number $$p$$, the following two things are true:


 * 1) The $$p$$-Sylow subgroup of $$G_1$$ is isomorphic to the $$p$$-Sylow subgroup of $$G_2$$. Note that there may be more than one $$p$$-Sylow subgroup on each side, but by Sylow's theorem, they are conjugate, hence the criterion does not depend on the choice of Sylow subgroup.
 * 2) The fusion system induced by $$G_1$$ on its $$p$$-Sylow subgroup is isomorphic (as a saturated fusion system) to the fusion system induced by $$G_2$$ on its $$p$$-Sylow subgroup, where we talk of this "isomorphic" after identifying the two Sylow subgroups using any isomorphism that exists by condition (1).

Facts

 * Any two fusion system-equivalent groups have the same order.
 * Finite nilpotent group implies every fusion system-equivalent group is isomorphic
 * Fusion system-equivalence preserves perfectness