Fusion systems for quaternion group

This article discusses possible fusion systems for the quaternion group.

Sylow subgroups realizing this fusion system
This fusion system is realized by a group having quaternion group as its 2-Sylow subgroup if and if it possesses a normal complement, so the 2-Sylow subgroup is a retract of the group and the group is a semidirect product of a normal $$p'$$-subgroup and the quaternion group, or equivalently the group is a 2-nilpotent group.

Some examples are below:

Sylow subgroups realizing this fusion system
Any situation where quaternion group arises as a 2-Sylow subgroup that is not a retract, i.e., does not have a normal complement. Any such example must admit special linear group:SL(2,3) as a subquotient. Examples are given below: