  We examine the possibilities of accommodating the muon $g-2$ anomaly reported
by Fermilab in the 2HDM with a discrete $Z_4$ symmetry in which an inert Higgs
doublet field ($H,~A,~H^\pm$) has the lepton flavor violation $\mu$-$\tau$
interactions. We study the case of light $H$ (5 GeV $<m_H<$ 115 GeV) and assume
the Yukawa matrices to be real and symmetrical. Considering relevant
theoretical and experimental constraints, especially for the multi-lepton
searches at the LHC, we find the muon $g-2$ anomaly can be explained within
$2\sigma$ range in the region of 5 GeV $<m_H<20$ GeV, 130 GeV $<
m_A~(m_{H^\pm})<$ 610 GeV, and 0.005 $<\rho<$ 0.014. Meanwhile, the
$\chi^2_\tau$ fitting the data of lepton flavour universality in the $\tau$
decays approaches to the SM prediction.
