Model structure on commutative monoids

David White's 2017 paper "Model structures on commutative monoids in general model categories" gives sufficient (but IIRC not necessary) conditions on a monoidal model category $\mathcal{C}$ so that $\mathcal{C}$ right-induces a model structure on the category of commutative monoids in $\mathcal{C}$. It seems like this is exactly what you're asking for, unless you've already checked White's paper and saw that your monoidal model category of interest does not satisfy White's conditions.

White's paper is available here: https://doi.org/10.1016/j.jpaa.2017.03.001 Arxiv version: https://arxiv.org/abs/1403.6759

Hadrian's other question brought me out of MathOverflow retirement (for the moment), so let me add an answer to this question. First, the condition in my paper Model structures on commutative monoids in general model categories is sufficient but not necessary. If you want a necessary and sufficient condition, you should look at the paper A necessary and sufficient condition for induced model structures by Kathryn Hess, Magdalena Kędziorek, Emily Riehl and Brooke Shipley.

The main reason I asked $f^{\square n}/\Sigma_n$ to be a trivial cofibration (whenever $f$ is one) was so that transfinite compositions of pushouts of such maps would be weak equivalences. So, you could immediately be more general than "Model structures on commutative monoids in general model categories" by asking only that $(-)^{\square n}/\Sigma_n$ takes trivial cofibrations into a class of maps that is closed under transfinite compositions of pushouts and contained in the weak equivalences. And, I did formulate a transfer result in that generality in Remark 6.1.3 of Bousfield Localization and Algebras over Colored Operads, joint with Donald Yau. That extra generality was useful for proving that all colored operads are admissible in the positive model structure on symmetric spectra (and the class of maps in question are the trivial cofibrations in the injective model structure).

I also want to point out that this is not the only way to right induce model structures. The technique here involves free algebra extensions, i.e., pushouts of morphisms $F(A) \to F(B)$, where $F$ is the free algebra functor (for commutative monoids $F(X) = Sym(X) = 1 \coprod X \coprod (X^{\otimes 2}/\Sigma_2) \coprod (X^{\otimes 3}/\Sigma_3) \coprod \dots$). An alternative way to right induce model structures is to run the path object argument in the category of algebras. This approach is spelled out in Berger and Moerdijk's:

https://ems.press/content/serial-article-files/42984

Specifically, see section 2.6, Theorem 3.1, and Theorem 3.2. The case Hadrian is interested in is a cartesian monoidal model structure, which makes things easier. The problem is that I could not figure out if Verity's model structure on stratified simplicial sets (a.k.a. the J-complicial model structure) has a symmetric monoidal fibrant replacement functor. If it does, then Hadrian could apply Proposition 4.1 of Berger-Moerdijk and conclude that all operads are admissible in Strat (or, Theorem 5.7 of this paper of mine with Donald Yau to conclude that all colored operads are admissible). This would avoid the analysis of symmetric products of the answer I just posted and would also give you a more general conclusion.

Lastly, let me try to answer the question that was posed. It is not true that any simplicial combinatorial monoidal model category $M$ has a transferred (right induced) model structure on commutative monoids. A counterexample is the projective model structure on chain complexes over a field of characteristic 2. This is explained in Section 5.1 of my commutative monoids paper. To see that the model category is simplicial, please check out section 4.2 of this paper of mine with Boris Chorny. Note that this example was cut in a later revision (which also changed the title of that paper), but the argument is there on arXiv.