I have a smooth projective $k$-scheme $X$ with a local system $F$ (locally constant sheaf) of finite dimensional $k$-vector spaces (on étale topology). My question is whether there exists a finite étale morphism $f \colon Y \rightarrow X$ such that pullback of $F$ is a constant sheaf on $Y$.

I am still not familiar with the theory of étale fundamental groups, but if we work in traditional topology then there is a correspondence between local systems of $k$-vector spaces and $k$-vector space representations of $\pi_1(X,x)$. Moreover if $F$ correspond to $\phi \colon \pi_1(X, x) \rightarrow V$, then $f^{-1}F$ correspond to $\phi' \colon \pi_1(Y,y) \rightarrow \pi_1(X,x) \rightarrow V$.

Is there a similar correpondence for étale fundamental group?

From that $f^{-1}F$ is trivial if $Im(\phi')$ is trivial, i.e. $Im(\pi_1(Y,y)) \subset ker(\phi)$. In topology we can find a cover such that the image is exactly $ker(\phi)$.

Can we do the same for étale fundamental group? I suppose that since I am looking for finite étale morphism, I need the condition that $ker(\phi)$ has finite index.

étaletopology (for the Zariski topology a locally constant sheaf is constant). $\endgroup$ – abx Dec 5 '19 at 14:45