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

Cohomology of linking systems with twisted coefficients by a $p$-solvable action