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
$$
(\Omega^\infty X)_{hG} \to \Omega^\infty(X_{hG})\,?
$$