  In this paper we study the cohomology of the geometric realization of linking
systems with twisted coefficients. More precisely, given a prime $p$ and a
$p$-local finite group $(S,\mathcal{F},\mathcal{L})$, we compare the cohomology
of $\mathcal{L}$ with twisted coefficients with the submodule of
$\mathcal{F}^c$-stable elements in the cohomology of $S$. We start with the
particular case of constrained fusion systems. Then, we study the case of
$p$-solvable actions on the coefficients.
