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 M₃ that are obtained by a covering construction from a primitive Teichm\"uller curve in M₂. We describe the action on homology modulo two of the affine groups of translation surfaces generating these primitive curves. We also classify the SL₂( Z)-orbits of square-tiled surfaces in H(2, 2) that are a cover of a genus two one.