A search for a Higgs boson decaying into a pair of electrons or muons and a photon is described. Higgs boson decays to a Z boson and a photon (H → Zγ → ℓℓγ, ℓ = e or μ), or to two photons, one of which has an internal conversion into a muon pair (H → γ$^{}$γ → μμγ) were considered. The analysis is performed using a data set recorded by the CMS experiment at the LHC from proton-proton collisions at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{−1}$. No significant excess above the background prediction has been found. Limits are set on the cross section for a standard model Higgs boson decaying to opposite-sign electron or muon pairs and a photon. The observed limits on cross section times the corresponding branching fractions vary between 1.4 and 4.0 (6.1 and 11.4) times the standard model cross section for H → γ$^{}$γ → μμγ (H → Zγ → ℓℓγ) in the 120–130 GeV mass range of the ℓℓγ system. The H → γ$^{*}$γ → μμγ and H → Zγ →ℓℓγ analyses are combined for m$_{H}$ =125GeV, obtaining an observed (expected) 95% confidence level upper limit of 3.9 (2.0) times the standard model cross section.