This paper presents a search for Higgs bosons decaying to four leptons, either electrons or muons, via one or two light exotic gauge bosons ${Z}_{d}$, $H\ensuremath{\rightarrow}Z{Z}_{d}\ensuremath{\rightarrow}4\ensuremath{\ell}$ or $H\ensuremath{\rightarrow}{Z}_{d}{Z}_{d}\ensuremath{\rightarrow}4\ensuremath{\ell}$. The search was performed using $pp$ collision data corresponding to an integrated luminosity of about $20\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at the center-of-mass energy of $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$ recorded with the ATLAS detector at the Large Hadron Collider. The observed data are well described by the Standard Model prediction. Upper bounds on the branching ratio of $H\ensuremath{\rightarrow}Z{Z}_{d}\ensuremath{\rightarrow}4\ensuremath{\ell}$ and on the kinetic mixing parameter between the ${Z}_{d}$ and the Standard Model hypercharge gauge boson are set in the range $(1--9)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}5}$ and $(4--17)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}2}$ respectively, at 95% confidence level assuming the Standard Model branching ratio of $H\ensuremath{\rightarrow}Z{Z}^{*}\ensuremath{\rightarrow}4\ensuremath{\ell}$, for ${Z}_{d}$ masses between 15 and 55 GeV. Upper bounds on the effective mass mixing parameter between the $Z$ and the ${Z}_{d}$ are also set using the branching ratio limits in the $H\ensuremath{\rightarrow}Z{Z}_{d}\ensuremath{\rightarrow}4\ensuremath{\ell}$ search, and are in the range $(1.5--8.7)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}4}$ for $15<{m}_{{Z}_{d}}<35\text{ }\text{ }\mathrm{GeV}$. Upper bounds on the branching ratio of $H\ensuremath{\rightarrow}{Z}_{d}{Z}_{d}\ensuremath{\rightarrow}4\ensuremath{\ell}$ and on the Higgs portal coupling parameter, controlling the strength of the coupling of the Higgs boson to dark vector bosons are set in the range $(2--3)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}5}$ and $(1--10)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}4}$ respectively, at 95% confidence level assuming the Standard Model Higgs boson production cross sections, for ${Z}_{d}$ masses between 15 and 60 GeV.