$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}$Recall that $n$-simplices of the classifying space (simplicial set) $BG$ of a group are orbits of the diagonal action by shifts 
on $n+1$-tuples of its elements. But what if we take elements from some other set the group acts on ? 

That is, the same definition can be made for an arbitrary action of a group on a set,
e.g. the Galois action on $\Bbb{\bar Q}$, or a symmetric group acting by permutations. Note that for the trivial action
of a group on a singleton I get the trivial simplicial object, and thus this is not the Borel construction. 
Where can I read about this ? What is the right terminology ? 

In notation: let $G$ be a group acting on a set $X$, i.e. we have $\tau:G\rightarrow \Aut X$. 
Recall that $B(G)_n = G^{n+1}/G$, and I define 
$$B'(G \xrightarrow{\tau} \Aut X):=X^{n+1}/G$$ 
Note that for $X=G$ and $\tau:G\rightarrow \Aut G$  the action by shifts,
$B(G)=B'(\tau)$, and for the trivial action of a group on a singleton, 
$B'(G\rightarrow \Aut(\{\mathrm{pt}\}))$ is trivial as well.

What what can be said about $B'(\tau)$ and where can I read about it ? 
In particular, for $G=\Gal( \Bbb{\bar Q/ Q})$ action on $\mathbb{\bar Q}$, or perhaps
$\Gal(K/\Bbb{Q})$ action on $K$ for a number field ? Has this been considered in number theory ?
Is this construction trivial for some reason ?   

For the permutation representation of an infinite symmetric group one gets a simplicial set representing equivalence relations;
for the finite symmetric group $S_n$ it classifies equivalence relations with at most $n$ equivalence classes.

I am also interested in the following "ordered" modification of the construction for the group $G=\Aut(\Bbb{Q}^\leq)$ 
of automorphisms of a dense linear order $\Bbb{ Q}$: an $n$-simplex is an orbit of the action on _non-decreasing_ $n+1$-tuples.
This simplicial set has the property that in each dimension there is a unique non-degenerate simplex, and, moreover,
all its faces are non-degenerate, and is weakly contractible, according to [answers here](https://mathoverflow.net/questions/439687/the-simplicial-set-with-a-unique-non-degenerate-simplex-in-each-dimension).If you truncate the simplicial set, for odd dimensions you get a sphere up to homotopy, and for even 
dimensions something weakly contractible, e.g. for $n=2$ the dunce hat.