TITLE: Homology of homotopy fixed point spaces
QUESTION [7 upvotes]: This is a general question for the homotopy theory crowd: How does one go about computing the homology and homotopy groups of homotopy fixed point spaces $X^{hG}:= Map^G(EG, X)$ for the action of a group $G$ on a space $X$?  There seem to be some tools:

Lannes' theory: which allows you to compute (or at least say something about) $H_*(X^{hG}, \mathbb{F}_p)$ when $G$ is a $p$-group.
Homotopy fixed point spectral sequences, which allow you to compute the stable homotopy groups of homotopy fixed point spectra.

Are there other tools out there?  I feel like (1.) should be the harder version of a fact that I'm missing about computing $H_*(X^{hG}, \mathbb{F}_p)$ when $|G|$ is coprime to $p$.  Regarding (2.), is there any hope of an unstable homotopy fixed point spectral sequence?

REPLY [4 votes]: Hej Craig,
Re (2) as Tilman says in his comment, there is an unstable homotopy fixed point spectral sequence, a special case of the spectral sequence of a homotopy limit as described by Bousfield and others.
Re (1) when X is finite (and more generally), Lannes theory should be seen as generalization of ordinary Smith theory. Smith theory only works for p-groups, so I don't think it is a harder version of a prime-to-p statement.