Echoes in genus three of Teichm\"uller curves in genus two

We classify the Teichm\"uller curves in the moduli space of genus three Riemann surfaces $\mathcal M_3$ that are obtained by a covering construction from a primitive Teichm\"uller 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 $\mathrm{SL}_2(\mathbb Z)$-orbits of square-tiled surfaces in $\mathcal H(2, 2)$ that are a cover of a genus two one.