Let $H$ be a subgroup of $G$ a compact Lie group and let $\text{spectra}[G]$ be the category of naive $G$-spectra (ie G-objects in the category of spectra). Then there is a forgetful functor $i^*$ from $\text{spectra}[G]$ to $\text{spectra}[H]$ with a left adjoint $G_+ \wedge_H -$.

**What can be said about the adjunction $(G_+ \wedge_H - , i^*)$ on homotopy categories?**

For example:

- I believe the forgetful functor $i^*$ to be faithful, does anyone have a reference?
- Is the left adjoint also faithful?
- I've been told this adjunction (probably) satisfies some form of homotopy descent. What does this tell me? (Presumably that some spectral sequence converges.)
- Is every $O(n)$-spectrum in the image of the left adjoint (on homotopy categories)?

(The specific case I'm interested in is $H$ is the symmetric group on $n$ letters and $G$ is the orthogonal group $O(n)$.)