The Barratt-Eccles operad is an operad in simplicial sets
that provides a particularly nice model for the $E
_∞$-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 $Σ(n)×(Σ(a_1)×⋯×Σ(a_n))→Σ(a_1+⋯+a_n)$
is given by stacking the permutations in $Σ(a_1)$ together and composing them
with the block permutation in $Σ(a_1+⋯+a_n)$ 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 the $E_\infty$-operad.
Specifically, I am interested in the
$E_\infty$-group operad, which models homotopy coherent commutative groups,
for example, algebras in spaces over this operad are a model for connective spectra;
it is obtained from the $E_\infty$-operad by adding homotopy coherent inverses.
Another interesting case is the $E_\infty$-ring operad,
which models homotopy coherent commutative rings,
for example, algebras in spaces over this operad
are a model for connective $E_\infty$-ring spectra;
it is obtained from the $E_\infty$-group operad by adding
a homotopy coherent commutative multiplication operation
with the appropriately coherent distributive property.

**Is there an analog of the Barratt-Eccles construction for the $E_\infty$-group and $E_\infty$-ring operads?**