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In this paper we give a definition of (centric) Mackey functor over a fusionsystem which generalizes the notion of Mackey functor over a group. In thiscontext we prove that, given some conditions on a related ring, the centricBurnside ring over a fusion system (as defined by Diaz and Libman) acts on anycentric Mackey functor. We also prove that the Green correspondence holds forcentric Mackey functors over fusion systems. As a means to prove this weintroduce a notion of relative projectivity for centric Mackey functors overfusion systems and provide a decomposition of a particular product in$\mathcal{O}\left(\mathcal{F}^c\right)_{\sqcup}$ in terms of the product in$\mathcal{O}\left(N_{\mathcal{F}}\left(H\right)^c\right)_{\sqcup}$.

Green correspondence on centric Mackey functor over fusion systems