We can via the bar construction canonically associate to a monoid $A$ 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 induced from  two-sided  [bar construction][1] $B(1,A,1)$ 
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 shall 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 are constructed is formally 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 finer 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 generalization 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 in same manner as we 
did for bar construction and Cech nerve, how the obtained simplicial
object $\overline{BG}$ would look like and how is it related to "classical" 
$BG$? 

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


  [1]: https://ncatlab.org/nlab/show/bar+construction