Let G be a finite group and X a topos with homotopy coherent G-action. From this, we construct a stable homotopy theory SpG(X) which recovers and extends the theory of genuine G-spectra. We explain what our construction yields when: (i) X is the topos of sheaves on a topological space with G-action (ii) X is the etale C2-topos of a scheme S adjoined a square root of -1.