A search for a light charged Higgs boson ($\mathrm{H}^{+}$) decaying to a W boson and a CP-odd Higgs boson (A) in final states with $\mathrm{e}\mu\mu$ or $\mu\mu\mu$ is performed using data from pp collisions at $\sqrt{s} =$ 13 TeV, recorded by the CMS detector at the LHC and corresponding to an integrated luminosity of 35.9 fb$^{-1}$. In this search, it is assumed that the $\mathrm{H}^{+}$ boson is produced in decays of top quarks, and the A boson decays to two oppositely charged muons. The presence of signals for $\mathrm{H}^{+}$ boson masses between 100 and 160 GeV and A boson masses between 15 and 75 GeV is investigated. No evidence for the production of the $\mathrm{H}^{+}$ boson is found. Assuming branching fractions ${\mathcal{B}({\mathrm{H}^{+}\to\mathrm{W^{+}}\mathrm{A} })}=$ 1 and ${\mathcal{B}({\mathrm{A} }\to\mu^{+}\mu^{-})}=3\times10^{-4}$, upper limits at 95% confidence level on the branching fraction of the top quark, ${\mathcal{B}({\mathrm{t}\to\mathrm{b}\mathrm{H}^{+}})}$, of 0.63 to 2.9% are obtained, depending on the masses of the $\mathrm{H}^{+}$ and A bosons. These are the first limits on ${\mathcal{B}({\mathrm{t}\to\mathrm{b}\mathrm{H}^{+}})}$ in the decay mode of the $\mathrm{H}^{+}$ boson: ${\mathrm{H}^{+}\to\mathrm{W^{+}}\mathrm{A} }\to\mathrm{W^{+}}\mu^{+}\mu^{-}$.

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