  We build a spectral sequence converging to the cohomology of a fusion system
with a strongly closed subgroup. This spectral sequence is related to the
Lyndon-Hochschild-Serre spectral sequence and coincides with it for the case of
an extension of groups. Nevertheless, the new spectral sequence applies to more
general situations like finite simple groups with a strongly closed subgroup
and exotic fusion systems with a strongly closed subgroup. We prove an analogue
of a result of Stallings in the context of fusion preserving homomorphisms and
deduce Tate's p-nilpotency criterion as a corollary.
