Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups

Let G be a closed subgroup of the semi-direct product of the nth Morava stabilizer group Sn with the Galois group of the field extension Fpn/Fp. We construct a “homotopy fixed point spectrum” EnhG whose homotopy fixed point spectral sequence involves the continuous cohomology of G. These spectra have the expected functorial properties and agree with the Hopkins-Miller fixed point spectra when G is finite.