  For a profinite group $G$, we define an $S[[G]]$-module to be a certain type
of $G$-spectrum $X$ built from an inverse system $\{X_i\}_i$ of $G$-spectra,
with each $X_i$ naturally a $G/N_i$-spectrum, where $N_i$ is an open normal
subgroup and $G \cong \lim_i G/N_i$. We define the homotopy orbit spectrum
$X_{hG}$ and its homotopy orbit spectral sequence. We give results about when
its $E_2$-term satisfies $E_2^{p,q} \cong \lim_i H_p(G/N_i, \pi_q(X_i))$. Our
main result is that this occurs if $\{\pi_\ast(X_i)\}_i$ degreewise consists of
compact Hausdorff abelian groups and continuous homomorphisms, with each
$G/N_i$ acting continuously on $\pi_q(X_i)$ for all $q$. If $\pi_q(X_i)$ is
additionally always profinite, then the $E_2$-term is the continuous homology
of $G$ with coefficients in the graded profinite
$\widehat{\mathbb{Z}}[[G]]$-module $\pi_\ast(X)$. Other results include
theorems about Eilenberg-Mac Lane spectra and about when homotopy orbits
preserve weak equivalences.
