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Two Higgs-doublet models usually consider an ad-hoc $\mathbb{Z}_2$ discretesymmetry to avoid flavor changing neutral currents. We consider a new class oftwo Higgs-doublet models where $\mathbb{Z}_2$ is enlarged to the symmetry group${\cal{F}}\rtimes \mathbb{Z}_2$, i.e. an inner semi-direct product of adiscrete symmetry group ${\cal{F}}$ and $\mathbb{Z}_2$. In such a scenario thesymmetry constrains the Yukawa interactions but goes unnoticed by the scalarsector. In the most minimal scenario, $\mathbb{Z}_3 \rtimes \mathbb{Z}_2 =D_3$, flavor changing neutral currents mediated by scalars are absent at treeand one-loop level, while at the same time predictions to quark and leptonmixing are obtained, namely a trivial CKM matrix and a PMNS matrix (uponintroduction of three heavy right-handed neutrinos) containing maximalatmospheric mixing. Small extensions allow to fully reproduce mixingparameters, including cobimaximal mixing in the lepton sector (maximalatmospheric mixing and a maximal $CP$ phase).

Two-Higgs-Doublet Models with a Flavored $\mathbb{Z}_2$