Let $M_g$ be the moduli space of genus $g$ curves, $A_g$ be the moduli space of principally polarized dimension $g$ abelian varieties. They have dimensions $3g-3,g(g+1)/2$ respectively. The Torelli map is a map $\tau_g: M_g \to A_g$ that takes a curve to it's Jacobian.

For $g=3$, $M_g$ and $A_g$ have the same dimension $6$ and I have heard that $\tau_3$ is a 2-cover that is ramified exactly at the hyperelliptic locus. So there should be an automorphism of $\tau_3$ that fixes the hyperelliptic locus, right? Can we give an explicit description at the level of genus 3 curves? That is, something like: Take some canonical map $C \to X$ for some variety $X$ and pull back by some involution on $X$?

(I would also appreciate a reference for the fact that $\tau_3$ is a double cover, it should have something to do with the involution on abelian varieties not extending to a general curve.)