The order of the torsion group does only depend on the choice of the line in the complete intersection of the quadratic relations Q in ℙ11. A numerical Godeaux surface X with torsion group ℤ/4 has two special bicanonical curves: one reducible curve of the form D1+D3, where Di ∈|KX + ti| with a torsion element ti, and a double curve 2D2, where D2 ∈|KX + t2|. To construct surfaces with such curves, the chosen line in Q must intersect two different loci in Q. We choose two different components of these loci and evaluate the condition that a line through two general points is completely contained in the variety Q. The result is a list of pairs of ideals such that each line through two general points is completely contained in Q and intersect the corresponding loci in a point.