  For a profinite group, we construct a model structure on profinite spaces and
profinite spectra with a continuous action. This yields descent spectral
sequences for the homotopy groups of homotopy fixed point space and for stable
homotopy groups of homotopy orbit spaces. Our main example is the Galois action
on profinite \'etale topological types of varieties over a field. One
motivation is to understand Grothendieck's section conjecture in terms of
homotopy fixed points.
