gives rise to a value of $\theta^l_{13}$ as indicated by the latest experimental results. An operator is induced with the help of the field $\eta$ which gives rise to a contribution to the neutrino mass matrix that breaks one of the two $Z_2$ symmetries responsible for TB mixing in the neutrino sector, while tri-maximal (neutrino) mixing is still protected by the intact $Z_2$ symmetry. The broken $Z_2$ symmetry is identified with $\mu -\tau$ symmetry, and thus it becomes possible to generate $\theta_{13}^l$ of the correct size. At the same time, one encounters a deviation of the atmospheric mixing angle from its maximal value which is proportional to $\theta_{13}^l$. The introduction of $\eta$ does not affect the successful predictions of the original $S_4 \times SU(5)$ model. In particular, the Gatto-Sartori-Tonin (GST) relation [@GST], the size of the quark mixing angles $\theta_{13}^q$ and $\theta_{23}^q$, and the Georgi-Jarlskog (GJ) relations [@GJ] are unaltered. Also the leading correction to the solar mixing angle remains the same, whereas the atmospheric mixing angle, as explained, receives larger corrections. As will be shown, the deviations of both these angles from TB mixing are subject to sum rule bounds. Furthermore, we discuss a simple ultraviolet (UV) completion of 