2-Fusion Systems with Components of Type SL2(q)

We study a problem in fusion systems designed to mimic, simplify, and generalize parts of the Classification of Finite Simple Groups. A 2-fusion system is a 2-group S with a family of injective homomorphisms on subgroups of S. Fusion systems arise in the study of modular representations and classifying spaces, and our results have potential ramifications beyond finite group theory. One problem is to determine S or, whenever possible, the entire 2-fusion system from the knowledge of certain subgroups and maps. We consider the case where S contains a subgroup and maps that arise in the Classification with standard component of type SL₂(q).