  A systematic study of one-loop contributions to the decay channels
$h\rightarrow l\bar{l}\gamma$ with $l=\nu_{e,\mu, \tau}, e, \mu$, performed in
Higgs extended versions of the Standard Model, is presented in the 't
Hooft-Veltman gauge. Analytic formulas for one-loop form factors are expressed
in terms of the logarithm and di-logarithmic functions. As a result, these form
factors can be reduced to those relating to the loop-induced decay processes
$h\rightarrow \gamma\gamma, Z\gamma$, confirming not only previous results
using different approaches but also close relations between the three kinds of
the loop-induced Higgs decay rates. For phenomenological study, we focus on the
two observables, namely the enhancement factors defined as ratios of the decay
rates calculated between the Higgs extended versions and the standard model,
and the forward-backward asymmetries of fermions, which can be used to search
for Higgs extensions of the SM. We show that direct effects of mixing between
neutral Higgs bosons and indirect contributions of charged Higg boson exchanges
can be probed at future colliders.
