Let $X$ be an (naive) $O(n)$-spectrum (I'm choosing to work with orthogonal spectra). I've recently come across the following results,
$$(S^{n-1} \wedge X)_{hO(n)} \simeq X_{hO(n-1)}$$
and
$$\Omega^\infty (X_{hG)}) \simeq (\Omega^\infty X)_{hG} $$
where $G$ is a compact Lie group.
I've spent the last few days trying to find sources for these results - and trying to prove them myself to little avail.
Any help/suggestions or references would be greatly appreciated.
Added The answer by Tyler has shown that the second is false. This raises the question about how close can we get, in the following sense: If $X$ is $n$-connected, then how connected is the map $$ (\Omega^\infty X)_{hG} \to \Omega^\infty(X_{hG})\,? $$