Let $E$ be a spectrum acted upon a finite group $G$. Is there a general way of computing the homology of the homotopy fixed point spectrum $E^{hG}$ in terms of that of $E$? (I'm aware that there is a spectral sequence for computing $\pi\_\* E^{hG}$ in terms of $\pi\_\* E$, but smashing with some other spectrum probably doesn't preserve homotopy fixed points.)


Here's a specific example I have in mind. Take connective $K$-theory $bu$. This has an action of $\mathbb{Z}/2$, which comes from the $\mathbb{Z}/2$-action on $K$-theory (given on the level of cohomology theories by complex conjugation of vector bundles). ~~Then $bu^{\mathbb{Z}/2} = bo$.~~ Actually, this is only true before taking connective covers. 


Let's say I know how to compute the mod 2 homology of $bu$ (it's $\mathbb{Z}/2[\zeta\_1^2, \zeta\_2^2, \zeta\_3, \zeta\_4, \dots]$ as a comodule over the dual Steenrod algebra). Does that give any information about $H\_\*(bo; \mathbb{Z}/2)$? Is there a good reference for this material? 



Homology of homotopy fixed point spectra
Consider $G$ of order $2$ acting trivially on the sphere spectrum $S^0$. In this key example, smashing with $H\mathbb Z/2$ drastically fails to commute with $^{hG}$: If it did commute, then the mod $2$ homology of the homotopy fixed point spectrum would be the homotopy of $(H\mathbb Z/2)^{hG}$, so $\mathbb Z/2$ in nonpositive dimensions and $0$ in positive dimensions. But the (solved) Segal conjecture shows that the homotopy fixed point spectrum is connective.


