TITLE: Reference for iterated homotopy fixed points?
QUESTION [6 upvotes]: What are (good) references for results about iterated homotopy fixed points?  That is, suppose G is a topological group acting on a space (or spectrum) X, and H is a normal subgroup of G.  Then one would like to first compute the homotopy fixed points of X with respect to H, and use that as a stepping stone to compute the homotopy fixed points of X with respect to G.
(I am independently interested in both the space and spectrum versions, so am happy with pointers, comments regarding either.)

REPLY [8 votes]: The statement XhG = (XhH)hG/H is true for any G-object X of any complete (∞,1)-category C.  An object of C with a G-action is the same as a functor BG → C where BG represents the category (or (∞,1)-category if G is not discrete) with a single object with automorphism group G.  The G-fixed points are the homotopy limit of this functor, or equivalently its right Kan extension along the functor BG → •.  We can factor this latter functor as p: BG → B(G/H) followed by q: B(G/H) → •.  So
$X^{hG} = (qp)_* X = q_* p_* X = (p_* X)^{hG/H}$
It remains to compute the right Kan extension of X along p.  On the object • of B(G/H), it is given as the limit of the diagram X over the category • ↓ G, which is the translation groupoid of G acting on G/H, or equivalently BH.  So indeed $p_* X = X^{hH}$.  Identifying the action of G/H is left as an exercise for the reader. :)