Small essential 2-subgroups in fusion systems

A (saturated) fusion system on a p-group P contains data about conjugacy within P, the typical case being the system induced by a group on its Sylow p-subgroup. Fusion systems are completely determined by looking at their essential subgroups, which must admit an automorphism of order coprime to p. For p=2, we describe two new methods that address the question: given an essential subgroup $E<P$ of a fusion system on P, what can we say about P? In particular, one method gives us sufficient conditions to deduce that $E\triangleleft P$, while the other explores cases where we have strong control over the normaliser tower of E in P.