  In this paper we consider a Higgs boson with mass and other properties
compatible with those of the recently discovered Higgs particle at the LHC, and
explore the possibility of new Higgs leptonic decays, beyond the standard
model, with the singular feature of being lepton flavor violating (LFV). We
study these LFV Higgs decays, $H \to l_k\bar l_m$, within the context of the
inverse seesaw model (ISS) and consider the most generic case where three
additional pairs of massive right-handed singlet neutrinos are added to the
standard model particle content. We require in addition that the input
parameters of this ISS model are compatible with the present neutrino data and
other constraints, like perturbativity of the neutrino Yukawa couplings. We
present a full one-loop computation of the BR($H \to l_k\bar l_m$) rates for
the three possible channels, $l_k\bar l_m=\mu \bar \tau,\, e \bar \tau,\, e
\bar \mu$, and analyze in full detail the predictions as functions of the
various relevant ISS parameters. We study in parallel the correlated one-loop
predictions for the radiative decays, $l_m \to l_k \gamma$, within this same
ISS context, and require full compatibility of our predictions with the present
experimental bounds for the three radiative decays, $\mu \to e \gamma$, $\tau
\to \mu \gamma$, and $\tau \to e \gamma$. After exploring the ISS parameter
space we conclude on the maximum allowed LFV Higgs decay rates within the ISS.
