The Barratt-Eccles operad is an operad in simplicial sets that provides a particularly nice model of an E∞-operad; algebras in spaces over the Barratt-Eccles operad model E∞-spaces, i.e., homotopy coherent commutative monoids in spaces. It can be described concretely by applying the nerve functor componentwise to an operad in groupoids, which itself is obtained by applying the codiscrete groupoid functor componentwise to an operad Σ in sets such that Σ(n) is the symmetric group of order n and the operadic composition Σ(n)×(Σ(a₁)×⋯×Σ(aₙ))→Σ(a₁+⋯+aₙ) is given by stacking the permutations in Σ(aᵢ) together and composing them with the block permutation in Σ(a₁+⋯+aₙ) induced by the permutation in Σ(n). Here the codiscrete groupoid functor sends a set X to the groupoid with X as the set of objects and exactly one morphism between any pair of objects; it is the right adjoint to the forgetful functor from groupoids to sets that sends a groupoid to its underlying set of objects.
I am interested in similarly spirited constructions for various cousins of E∞-spaces.
Specifically, I am interested in group-like E∞-spaces, which can be thought of as homotopy coherent commutative groups and are a model for connective spectra.
Another interesting case is E∞-ring spaces, which can be thought of as homotopy coherent commutative rings, and are a model for connective E∞-ring spectra.
As pointed out by Peter May in his answer, operads cannot model such structures because they do not allow for operations with multiple outputs, e.g., diagonal maps, so a part of the question is what type of structure one should use.
For example, simplicial algebraic theories (see http://ncatlab.org/nlab/show/(%E2%88%9E,1)-algebraic+theory) seem to be a viable option. In particular, the Barratt-Eccles construction admits a particularly elegant formulation in terms of a (2,1)-algebraic theory (i.e., a groupoid-valued algebraic theory), see http://ncatlab.org/nlab/show/(2,1)-algebraic+theory+of+E-infinity+algebras.
Is there an analog of the Barratt-Eccles construction for group-like E∞-spaces and E∞-ring spaces?