Cohomology with twisted coefficients of the classifying space of a fusion system

We study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system FF. More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of FF-stable elements to FcFc-stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group. We then study the problem of inducing an idempotent from an FF-characteristic (S,S)(S,S)-biset and we show that, if the coefficient module is nilpotent, then the cohomology of the geometric realization of a linking system can be computed by FcFc-stable elements. As a corollary, we show that for any coefficient module, the cohomology of the classifying space of a p  -local finite group can be computed by these FcFc-stable elements.