Let X be a simplicial set. Its category of simplices, denoted by $\Delta(X)$, is the category whose objects are the pairs $(x,[n])$, with $x\in X_n$, and morphisms $\bar{c}:(y,[m])\to (x,[n])$, where $c:[m]\to [n]$ is a non-decreasing map and $c^{\ast}(y)=x$. In particular, if $X=NG$, the nerve of a group $G$, it is clear that there exists a canonical embedding $G\text{-Mod}\hookrightarrow \Delta(NG)\text{-Mod}$.
However, is (in the above sense) $\Delta(NG)\text{-Mod}$ larger than $G\text{-Mod}$? If so, can we characterize those $\Delta(NG)$-modules that come from $G$-modules?
