A search is presented for pairs of light pseudoscalar bosons, in the mass range between 4 and 15~\mathrm{GeV}, produced in decay of the 125 GeV Higgs boson. The decay mode where one light boson decays into $\tau$ leptons, while the other one decays into a pair of $\tau$ leptons or muons, is considered. The search is based on proton-proton collision data collected by the CMS experiment at a centre-of-mass energy of 13~\mathrm{TeV} and corresponding to an integrated luminosity of 35.9~\mathrm{fb}^{-1}. No significant excess of events above the standard model background expectation obtained from control regions in data is observed. The $95\%$ confidence level observed (expected) upper limits on the signal production cross section times branching fraction into the 4$\tau$ final state, relative to the standard model Higgs boson production cross section are set between 0.022 (0.027) and 0.23 (0.19) in the probed mass range.

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