  We study the cohomology with twisted coefficients of the geometric
realization of a linking system associated to a saturated fusion system
$\mathcal{F}$. More precisely, we extend a result due to Broto, Levi and Oliver
to twisted coefficients. We generalize the notion of $\mathcal{F}$-stable
elements to $\mathcal{F}^c$-stable elements in a setting of cohomology with
twisted coefficients by an action of the fundamental group.% or, in other word,
with locally constant coefficients. We then study the problem of inducing an
idempotent from an $\mathcal{F}$-characteristic $(S,S)$-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 $\mathcal{F}^c$-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 $\mathcal{F}^c$-stable elements.
