A search is described for a Higgs boson decaying into two photons, one of which has an internal conversion to a muon or an electron pair ($\ell\ell\gamma$). The analysis is performed using proton-proton collision data recorded with the CMS detector at the LHC at a centre-of-mass energy of 8 TeV, corresponding to an integrated luminosity of 19.7 fb$^{-1}$. The events selected have an opposite-sign muon or electron pair and a high transverse momentum photon. No excess above background has been found in the three-body invariant mass range $120 < m_{\ell\ell\gamma} < 150$ GeV, and limits have been derived for the Higgs boson production cross section times branching fraction for the decay $\mathrm{H} \to \gamma^* \gamma \to \ell \ell \gamma$, where the dilepton invariant mass is less than 20 GeV. For a Higgs boson with $m_{\mathrm{H}} =$ 125 GeV, a 95% confidence level (CL) exclusion observed (expected) limit is 7.7 (6.4$^{+3.1}_{-2.0}$) times the standard model prediction. Additionally, an upper limit at 95% CL on the branching fraction of $\mathrm{H} \to(\mathrm{J}/\psi)\gamma$ for the 125 GeV Higgs boson is set at 1.5$\times$10$^{-3}$.

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