Fermion masses and mixings, dark matter, leptogenesis and $g-2$ muon anomaly in an extended 2HDM with inverse seesaw

We propose a predictive $Q_4$ flavored 2HDM model, where the scalar sector is enlarged by the inclusion of several gauge singlet scalars and the fermion sector by the inclusion of right handed Majorana neutrinos. In our model, the $Q_4$ family symmetry is supplemented by several auxiliary cyclic symmetries, whose spontaneous breaking produces the observed pattern of SM charged fermion masses and quark mixing angles. The light active neutrino masses are generated from an inverse seesaw mechanism at one loop level thanks to a remnant preserved $Z_2$ symmetry. Our model succesfully reproduces the measured dark matter relic abundance only for masses of the DM candidate below $sim$ 0.8 TeV. Furthermore, our model is also consistent with the lepton and baryon asymmetries of the Universe as well as with the muon anomalous magnetic moment.