  A non-universal $U(1)_{X}$ extension to the Standard Model composed of two
scalar doublets and two scalar singlets together with three additional quark
singlets and two lepton singlets and three generations of right-handed and
Majorana neutrinos is made to explain lepton mass hierarchy, neutrino masses
via inverse seesaw mechanism and muon anomalous magnetic moment in an anomaly
free framework. In the present model, exotic and Standard Model particles
acquire mass thanks to vacuum expectation values at different scales, yet the
electron and the lightest neutrino are tree level massless but massive at
one-loop level. By considering a numerical exploration and under the constraint
of the Higgs mass, neutrino mass differences and PMNS matrix, it is found that
only contributions due to exotic neutrinos interacting with charged scalars are
relevant to muon $g-2$, though they are negative. Thus, the SUSY extension is
considered and it is found that muon $g-2$ can be explained by allowing
$U(1)_{X}$ vacuum expectation values to lie in the TeV scale thanks to SUSY
soft-breaking interactions for at least $\sim 10^{5}$ GeV masses. Thus, the
contribution due to exotic neutrinos interacting with $W$ gauge bosons is
positive and no longer negligible which added to all other contributions might
explain the anomaly.
