To a monoid $A$ the bar construction associates the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps from two-sided bar construction $B(1,A,1)$ of $A$. The constructed object looks like

$$ N(\mathbf{B}A)= \left( \cdots A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \stackrel{\to}{\to} {*} \right) $$

We know that if $A = G$ is a (finite, for sake of simplicity) group considered as category with one object and invertible morphisms, then $N(B A)$ or more precisely it's realization is the classifying space which we sheall call $BG$.

On the other hand it is known that if $C$ is a category with pullbacks and $U \to X$ is a morphism. More generally we can consider instead of fix $U$ the covers $U = \coprod_i U_i $ of a covering sieve $\{U_i \to X \}$. The Cech nerve $C(U)$ is the simplicial object in $C$ that in degree $k$ is given by the $(k+1)$-fold fiber product of $U$ over $X$ with itself :

$$ C(U) = \left( \cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \right) $$

The way the Cech nerve and the bar construction arise is the same and this lead me to following question. It is well know (or more precisely that's exacly the purpose) that this Cech nerve provides a canonical resolution for calculation of Cech cohomology of $X$.

But we know that generally the resolution in Cech nerve not provides a resolution for calculation of eg etale cohomology. The reason is that is simply not "fine" enough. Instead one consructs another resoluton using hypercovers of $X$. One can show that a resolution can be used to calculation of etale cohomology if it has certain coskeletal bahavior in each stage. nevertheless, although the construction of such hypercover is laborious, it is straigh forward, so we have our "cooking recipe". So we can consider the hypercover as natural, more powerful generation of Cech nerve.

My question is if we imitate literally the contruction of a hypercovering of scheme or stack $X$ on (let say finite) group $A:=G$ as above we did for bar construction and Cech nerve, how the obtained simplicial object $\overline{BG}$ would look like? How is it related to "classical" $BG$?

In other words if $BG$ from constructional point of view corresponds to the Cech nerve $C(U)$ of cover $U \to X$, is there an simplicial object (or respectively it's realization) $\overline{BG}$ which corresponds in analogous manner to a resolution by hypercover (over a scheme/stack)? Does this $\overline{BG}$ some nice properties, for example does it represent some interesting objects as in case of classical $BG$?