Control of fusion and cohomology of trivial source modules

Let G be a finite group and H a subgroup. We give an algebraic proof of Mislin's theorem which states that the restriction map from G to H on mod-p cohomology is an isomorphism if and only if H controls p-fusion in G. We follow the approach of P. Symonds [P. Symonds, Mackey functors and control of fusion, Bull. London Math. Soc. 36 (2004) 623–632] and consider the cohomology of trivial source modules.