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?