A characteristic subgroup for fusion systems

As a counterpart for the prime 2 to Glauberman's ZJ-theorem, Stellmacher proves that any nontrivial 2-group S has a nontrivial characteristic subgroup W(S) with the following property. For any finite Σ4-free group G, with S a Sylow 2-subgroup of G and with O2(G) self-centralizing, the subgroup W(S) is normal in G. We generalize Stellmacher's result to fusion systems. A similar construction of W(S) can be done for odd primes and gives rise to a Glauberman functor.