Echoes in genus three of Teichmüller curves in genus two

We classify the Teichmüller curves in the moduli space of genus three Riemann surfaces ${\mathcal{M}}_3$ that are obtained by a covering construction from a primitive Teichmüller curve in ${\mathcal{M}}_2$. We describe the action on homology modulo two of the affine groups of translation surfaces generating these primitive curves. We also classify the ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$-orbits of square-tiled surfaces in $\mathcal{H}(2,2)$ that are a cover of a genus two one.