  We consider a simple extension of the Standard Model (SM) with a vector-like
lepton and a local $U(1)'$ symmetry, motivated by the recent experimental
anomalies in the muon $g-2$ and the $W$ boson mass. The $U(1)'$ symmetry is
spontaneously broken by the VEVs of the dark Higgs scalar and the second Higgs
doublet, giving rise to the mixing between the muon and the vector-like lepton.
As a result, we obtain the desirable corrections to the muon $g-2$ and the $W$
boson mass simultaneously, dominantly due to the $Z'$ gauge interactions for
the vector-like lepton and the second Higgs doublet, respectively. We also
discuss the consistency of the model with the $Z$ boson decay width, the Higgs
couplings and the dilepton bounds.
