  We show that if an inclusion of finite groups H < G of index prime to p
induces a homeomorphism of mod p cohomology varieties, or equivalently an
F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd.
This generalizes classical results of Quillen who proved this when H is a Sylow
p-subgroup, and furthermore implies a hitherto difficult result of Mislin about
cohomology isomorphisms. For p=2 we give analogous results, at the cost of
replacing mod p cohomology with higher chromatic cohomology theories.
  The results are consequences of a general algebraic theorem we prove, that
says that isomorphisms between p-fusion systems over the same finite p-group
are detected on elementary abelian p-groups if p odd and abelian 2-groups of
exponent at most 4 if p=2.
