Let $X$ be a "reasonable" scheme (I am particularly interested in smooth algebraic varieties over a field). Let $Zar_X$ denote the (small) Zariski site of (open subschemes of) $X$ and $Et_X$ denote the (small) etale site of (etale schemes over) $X$. There is a natural map of sites $\pi:Et_X\to Zar_X$ (the direction of the site map is opposite to that of the functor on the categories of open sets).

Hence to any Zariski sheaf $A$ on $X$ one can assign its inverse image $\pi^*A$, which is an etale sheaf on $X$. There is also the adjoint functor $\pi_\ast$. It appears that $\pi_*\pi^*A=A$, so the functor $\pi^*$ is fully faithful. What is its essential image?

Given an etale sheaf $B$ on $X$, for any scheme point $i_x:x\to X$ there is the inverse image $i_x^*B$, which is an etale sheaf over $x$. Since $x$ is the spectrum of a field (namely, the residue field $k(x)$ of $X$ at $x$), an etale sheaf over $x$ can be viewed as a discrete module over the absolute Galois group $G_x$ of the field $k(x)$.

When the sheaf $B$ has the form $B=\pi^*A$, all the $G_x$-modules $i_x^*B$ are trivial (in the sense that the action of $G_x$ is trivial). Is the converse true?