The locus of lines in Q leading generically to ℤ/4-numerical Godeaux surfaces (with two fixed double base points of the bicanonical system) is birational to a union ℙ3 times ℙ1. The procedures gives two parametrization homomorphisms from Sa to the coordinate ring of a ℙ3 × ℙ1, where the image of each map gives a point in Q whose connecting line is completely contained in Q and leads generically to a ℤ/4-Godeaux surface.