The locus of lines in Q leading generically to ℤ/2-numerical Godeaux surfaces (with two fixed double base points of the bicanonical system) is birational to a bundle Z over ℙ3. The procedures gives two parametrization homomorphisms from Sa to the coordinate ring of Z, 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/2-Godeaux surface.