Given a comonad $\mathbf{G}=(G,\delta,\epsilon)$ on a category $A$, a natural transformation $t : G G\rightarrow G G$ is a symmetry of $\mathbf{G}$ if it sastisfies the quantum Yang-Baxter equation (transposed braid relation)
and also
In that case, the pair $(G,t)$ is called a symmetric comonad.
The simplicial set obtained by the standard construction from a symmetric comonad has always a structure of a symmetric simplicial set.
M. Grandis, Finite sets and symmetric simplicial sets, Theory of Applications of Categories, Vol. 8, No. 8, pp. 244–253, link (to come?).
Z. Škoda, Cyclic structures for simplicial objects from comonads, math.CT/0412001.