This paper presents a search for Higgs bosons decaying to four leptons, either electrons or muons, via one or two light exotic gauge bosons $Z_d$, $H\to Z Z_d \to 4\ell$ or $H\to Z_d Z_d \to 4\ell$. The search was performed using $pp$ collision data corresponding to an integrated luminosity of about 20 fb$^{-1}$ at the center-of-mass energy of $\sqrt{s}=8$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 $H\to Z Z_d \to 4\ell$ and on the kinetic mixing parameter between the $Z_d$ and the Standard Model hypercharge gauge boson are set in the range $(1$--$9)\times10^{-5}$ and $(4$--$17)\times10^{-2}$ respectively, at 95% confidence level assuming the Standard Model branching ratio of $H\to Z Z^* \to 4\ell$, 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 $H \to ZZ_d\to 4\ell$ search, and are in the range $(1.5$--$8.7)\times 10^{-4}$ for $15<m_{Z_d} < 35$ GeV. Upper bounds on the branching ratio of $H\rightarrow Z_dZ_d \to 4\ell$ and on Higgs portal coupling parameter, controlling the strength of the coupling of the Higgs boson to dark vector bosons, are set in the range $(2$--$3)\times10^{-5}$ and $(1$--$10)\times 10^{-4}$ respectively, at 95 % confidence level assuming the Standard Model Higgs boson production cross sections, for $Z_d$ masses between 15 and 60 GeV.

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