  Complementing and extending the Inventiones work of Benson, Grodal, Henke
[Group cohomology and control of p-fusion, Invent. Math. 197 (2014), 491--507]
we give criteria for a space to have cohomology (strongly) F-isomorphic in the
sense of Quillen to the stable elements. We extend results about groups models
to fusion systems over discrete $p$-toral groups and profinite groups and
provide various applications to the Kan-Thurston Theorem.
