Models for equivariant genuine commutative ring spectra

I proved in my thesis that $CMon(C_O(G))$ admits a transferred model structure, if you work with the positive or positive flat stable model structure on $C_O(G)$ (for the former, see Mandell-May, for the latter, see Stolz's thesis). I then proved that $CMon(C_O(G))$ is Quillen equivalent to $E_G-Alg(C_O(G))$, in the section on rectification. A reference is arXiv:1403.6759. I don't think you can use the Pavlov-Scholbach stuff for this. As I recall, they don't touch anything equivariant.

My work on preservation under localization, together with Mike Hill's example of a localization that destroys $E_G-Alg$ structure, provides a concrete example that $E-Alg$ and $E_G-Alg$ are not Quillen equivalent. A reference is arXiv:1404.5197. Mike's example is reproduced there as Example 5.7. I just realized I never uploaded the submitted version of this paper to arxiv, so the arxiv version still has a mistake in Theorem 5.9. It's not true that such localizations preserve genuine commutativity, precisely because $E-Alg$ is not equivalent to $CMon$ (Justin Noel emailed me about this ages ago; embarrassing that I forgot to update arxiv). What's true is that preservation for CMon is the same as for $E_G$-Alg.

I never studied $C_\Sigma$, but Mark Hovey and I had some vague plans to think about it if we ever had time. It seemed like the approach we took in arXiv:1312.3846 (by now, a better reference is probably the HHR appendix of the Kervaire paper) would make it easy to prove such an equivalence. If you plan to go down that route, feel free to email me and we can talk more.

For the questions asked here, there is no difference between orthogonal $G$-spectra, symmetric $G$-spectra of either $G$-spaces or $G$-sSets, or EKMM $G$-spectra. For the first two, the nonequivariant arguments in Mandell-May-Schwede-Shipley http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf generalize directly, once one corrects Lemma III.8.4 in Mandell-May http://www.math.uchicago.edu/~may/PAPERS/MMMFinal.pdf, as is done in HHR. That lemma should say that for positive cofibrant $G$-spectra $X$ (in any good category of $G$-spectra), the natural map $E(G,\Sigma_n)\wedge_G X^n \to X^n/\Sigma_n$ is a weak equivalence, where $E(G,\Sigma_n)$ is the universal principal $(G,\Sigma_n)$-bundle. This implies that the homotopy categories of $E_G$-algebras and of commutative monoids are equivalent for any genuine $E_{\infty}$ $G$-operad $E_G$. As David White says, his thesis improves these equivalences to Quillen equivalences of model categories. The "canonical" choice of $E_G$ is in the eyes of the beholder: there are many different natural choices with different applications of each: infinite Steiner or little discs $G$-operads, linear isometries $G$-operads, permutativity (or Barratt-Eccles) $G$-operads, etc. There are analogous equivalences using naive $E_{\infty}$-$G$-operads, which are nonequivariant $E_{\infty}$ operads regarded as $G$-trivial. That is the weakest in the hierarchy of point-set level kinds of genuine $G$-spectra with commutative ring structures given by Blumberg-Hill,whereas commutative monoids is the strongest.