We consider the extension of the Standard Model with an inert scalar doublet, three right-handed neutrinos, and singlet scalar fields, φ and S. In this model, neutrino masses are zero in the limit of the unbroken Z4 discrete symmetry. We show that when the singlet scalar field φ gets a VEV, the Z4
Nearby in the stack
symmetry is broken to
Z2
, and neutrino masses are generated at one-loops due to the mixings between the neutral components of the inert scalar doublet and the singlet scalar field
S
. There is a dark matter candidate from the lightest neutral scalar field, which is a mixture of the inert scalar doublet and the singlet scalar field
S
, in general. The
Z4
breaking mass terms are constrained by electroweak precision data and direct detection (DD) bounds for dark matter, favoring small mixings or almost degenerate masses for the DM scalars. As a result, we discuss the implications of the results for small neutrino masses and DD-safe dark matter.