A search for a light pseudoscalar Higgs boson (a) decaying from the 125 GeV (or a heavier) scalar Higgs boson (H) is performed using the 2016 LHC proton-proton collision data at $\sqrt{s} =$ 13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{-1}$, collected by the CMS experiment. The analysis considers gluon fusion and vector boson fusion production of the H, followed by the decay $\mathrm{H}\to\mathrm{a}\mathrm{a}\to\mu\mu\tau\tau$, and considers pseudoscalar masses in the range 3.6 $< {m_{\mathrm{a}}} <$ 21 GeV. Because of the large mass difference between the H and the a bosons and the small masses of the a boson decay products, both the $\mu\mu$ and the $\tau\tau$ pairs have high Lorentz boost and are collimated. The $\tau\tau$ reconstruction efficiency is increased by modifying the standard technique for hadronic $\tau$ lepton decay reconstruction to account for a nearby muon. No significant signal is observed. Model-independent limits are set at 95% confidence level, as a function of ${m_{\mathrm{a}}}$, on the branching fraction (${\mathcal{B}}$) for $\mathrm{H}\to\mathrm{a}\mathrm{a}\to\mu\mu\tau\tau$, down to 1.5 (2.0) $\times$ 10$^{-4}$ for ${m_{\mathrm{H}}} =$ 125 (300) GeV. Model-dependent limits on ${\mathcal{B}}(\mathrm{H}\to\mathrm{a}\mathrm{a})$ are set within the context of two Higgs doublets plus singlet models, with the most stringent results obtained for Type-III models. These results extend current LHC searches for heavier a bosons that decay to resolved lepton pairs and provide the first such bounds for an H boson with a mass above 125 GeV.

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