Can the Higgs field feel a dark force?

We argue that if an electroweak Higgs field possesses a dark gauge charge responsible for dark matter stability, the $W$-boson mass deviation is properly induced, besides appropriately generated neutrino masses. We examine a simple model in which the usual Higgs doublet plays the role but dark matter candidates are somewhat input by ad hoc. We look for a realistic model that fully realizes such observation, thereby neutrino mass and dark matter are naturally supplied by a dark non-abelian gauge symmetry.