The following question is an attempt at understanding various flavours of equivariant commutative ring spectra; it may not be suitable level for this forum.

Let $\mathcal{C}(G)$ be a symmetric monoidal *homotopical* category such that $Ho(\mathcal{C}(G))$ is the category $SH(G)$ of genuine $G$-equivariant spectra. Here $G$ is a finite group. The examples I have in mind are the category $\mathcal{C}_O(G)$ of $G$-objects in orthogonal spectra, with the G-equvariant stable weak equivalences (Schwede) or the category $\mathcal{C}_\Sigma(G)$ of T-symmetric spectra in the category of $G$-simplicial sets, with $T$ an appropriate simplicial version of the regular representation sphere (Mandell).

Going back to general $\mathcal{C}(G)$, we may then form the homotopical category $CMon(\mathcal{C}(G))$ of commutative, unital monoid objects in $\mathcal{C}(G)$ (with weak equivalences the underlying weak equivalences). Furthermore, given an operad $O$ with values in $\mathcal{C}(G)$ we can talk about the homotopical category $O-Alg(\mathcal{C}(G))$.

Note that $CMon(\mathcal{C}(G))$ is just $O-Alg(\mathcal{C}(G))$ for $O$ the commutative operad (all spaces just the tensor unit).

Any choice $\mathcal{C}(G)$ should admit a symmetric monoidal functor from the category $sSet(G)$ of $G$-simplicial sets. Then, given any operad in $G$-simplicial sets, we can talk about the induced operad in $\mathcal{C}(G)$. In particular we have "the" classical E-infinity operad $E \in Operads(sSet) \subset Operads(sSet(G))$ (consisting of spaces with trivial action). As far as I understand there is also the "genuine G-equivariant E-infinity operad" $E_G \in Operads(sSet(G))$, and in fact many operads between those two (see Blumberg-Hill).

So we now have six homotopical categories: $E-Alg(\mathcal{C}_O(G)), E_G-Alg(\mathcal{C}_O(G)), CMon(\mathcal{C}_O(G))$ and $E-Alg(\mathcal{C}_\Sigma(G)), E_G-Alg(\mathcal{C}_\Sigma(G)), CMon(\mathcal{C}_\Sigma(G))$. **My question is, which of these are known (or expected) to have equivalent homotopy categories?**

**Remark 1**

I'm phrasing this question in terms of homotopical categories because there are many different model structures, and there are many subtle issues regarding these, but as far as I can tell my question is not about this.

**Remark 2**

As far as I understand, if $G$ is the trivial group, all of these categories model commutative ring spectra and have equivalent homotopy categories.

**Remark 3**

If I understand correctly the work of Blumberg-Hill (and many others), then $Ho(E-Alg(\mathcal{C}_O(G))) \ne Ho(E_G-Alg(\mathcal{C}_O(G)))$ because the objects on the right have "norm maps" but on the left not. A related thing to say is that the homotopy groups on the right are tambara functors and on the left they have less structure.

I think it is also mentioned in loc. cit. that $Ho(E_G-Alg(\mathcal{C}_O(G))) = Ho(CMon(\mathcal{C}_O(G)))$.

**Remark 4**

There are various articles proving that $Ho(E-Alg(Spt(\mathcal{M},T))) = Ho(CMon(Spt(\mathcal{M},T)))$ for rather general model categories $\mathcal{M}$, see e.g. Pavlov-Scholbach. This seems to suggest to me that $Ho(E-Alg(\mathcal{C}_\Sigma(G))) = Ho(E_G-Alg(\mathcal{C}_\Sigma(G))) = Ho(CMon(\mathcal{C}_\Sigma(G)))$ (but checking the detailed list of requirements for their theorem is non-trivial). Note that this would be in stark contrast to the case of orthogonal spectra! So my best guess is that loc. cit. does not apply in our situation?

**Some references**

Schwede on equivariant orthogonal spectra: http://www.math.uni-bonn.de/people/schwede/equivariant.pdf

Mandell on equivariant symmetric spectra: http://pages.iu.edu/~mmandell/papers/gssfinal.dvi

Blumberg-Hill on equivariant E-infinity operads: https://arxiv.org/pdf/1309.1750v3.pdf

Pavlov-Scholbach: http://wwwmath.uni-muenster.de/sfb878/publications/files/phpimKZBl5582.pdf