Soft breaking of the $\mu\leftrightarrow \tau$ symmetry by $\mathbf{S}_{4}\otimes \mathbf{Z}_{2}$

The $\mu \leftrightarrow \tau$ symmetry has been ruled out by its predictions on the reactor and atmospheric angles, nevertheless, a breaking of this symmetry might provide correct values. For that reason, we build a non-renormalizable lepton model where the mixings arise from the spontaneous breaking of the $\mathbf{S}_{4}\otimes \mathbf{Z}_{2}$ discrete group, subsequently the $\mu \leftrightarrow \tau$ symmetry is broken in the effective neutrino mass matrix, that comes from the type II see-saw mechanism. As main result, the reactor and atmospheric angles are corrected and their values are in good agreement with the experimental data for the inverted hierarchy. Furthermore, we point out a link between the atmospheric angle and reactor one. In the quark sector, under certain assumptions, the generalized Fritzsch textures shape to the quark mass matrices so that the CKM matrix values are guaranteed.