This is a construction everybody talks about, but I have not seen it explicitly anywhere. Let $A$ be a finite group and $X$ be a scheme. An $A$-torsor $T$ on $X$ is a sheaf for the etale topology of $X$, such that it is locally constant with stalks $A$ on which $A$ acts by right translation.
Suppose also that there is a linear representation $A \xrightarrow{\rho} GL(V)$ into a finite dimensional $\overline{\mathbb{Q}_l}$-vector space $V$.
With these two data, how does one obtain a smooth $\overline{\mathbb{Q}_l}$-sheaf on $X$? I could not find an explicit construction anywhere, so either that or a reference where this is carried out in detail(in English, I hope) would be great. Thanks!
Perhaps what you mean is the following. In the following I assume that $A$ is constant and abelian. The first is (I believe) is clearly contextually implied by your post, and the second is just to avoid headache.
The $A$-torsors on $X$ are classified by the cohomology group $H^1_\text{et}(X,A)$. But, this is nothing more than $H^1_\text{cont.}(\pi_1^\text{et}(X),A)$ and, since $A$ is constant, this is nothing more than $\text{Hom}(\pi_1^\text{et}(X),A)$. In other words your $T$ gives rise to a continuous homomorphism $\pi_1^\text{et}(X)\to A$. Composing this with $\rho$ gives a continuous homomorphism $\pi_1^\text{et}\to\mathrm{GL}(V)$. As is well-known this then gives rise to a lisse $\overline{\mathbb{Q}}_\ell$-sheaf.
Does this answer your question?