  We study an Abelian gauge extension of the standard model with fermion
families having non-universal gauge charges. The gauge charges and scalar
content are chosen in such an anomaly-free way that only the third generation
fermions receive Dirac masses via renormalisable couplings with the Higgs
boson. Incorporating additional vector like fermions and scalars with
appropriate $U(1)$ charges can lead to radiative Dirac masses of first two
generations with neutral fermions going in the loop being dark matter
candidates. Focusing on radiative muon mass, we constrain the model from the
requirement of satisfying muon mass, recently measured muon anomalous magnetic
moment by the E989 experiment at Fermilab along with other experimental bounds
including the large hadron collider (LHC) limits. The anomalous Higgs coupling
to muon is constrained from the LHC measurements of Higgs to dimuon decay. The
singlet fermion dark matter phenomenology is discussed showing the importance
of both annihilation and coannihilation effects. Incorporating all bounds lead
to a constrained parameter space which can be probed at different experiments.
