Domestic intersections in fusion systems

Let F be a saturated fusion system over a p-group S and let E be a subsystem of F with AutE(S)=AutF(S). We prove that the fusion system F is generated by all E-morphisms and all F-automorphisms of some domestic intersections (called F∖E-domestic intersections in the paper) of F. We also get some results about control of fusion in finite groups as applications.