  We propose extended 3HDM and 4HDM models where the SM gauge symmetry is
enlarged by the spontaneously broken $S_4$ group, the preserved $Z_2$ and
broken $Z_4$ cyclic symmetries. Each model has three active $SU(2)$ scalar
doublets, in addition, the first one has an extra inert scalar singlet, whereas
the second model has an inert scalar doublet. Furthermore, each model has
several extra gauge singlet scalars, which are triplets under $S_4$. Both
models yield the same structure of the mass matrices for the fermion sector,
where a radiative seesaw generates the tiny light active neutrinos masses at
one-loop level, through the $S_4$ triplets. The presence of flavor changing
neutral currents mediated by heavy scalars allowed us to study the
$(K^0-\overline{K}^0)$ and $(B_{d,s}^0-\overline{B}_{d,s}^0)$ meson mixings, in
the parameter space that currently satisfies the experimental constraints. On
the other hand, due to the preserved $Z_2$ symmetry, our proposed models have
stable scalar and fermionic dark matter candidates. Furthermore, these models
are consistent with the current pattern of SM fermion masses and mixings, with
the measured dark matter relic abundance and successfully accommodate the
constraints arising from meson oscillations and oblique parameters. The extra
scalars in our models provide radiative corrections to the oblique parameters,
where due to the presence of the scalar inert doublet, renders the 4HDM less
restrictive than the 3HDM one.
