A search for charged Higgs boson decaying to a charm and a bottom quark (H$^+\to$ c$\overline{\mathrm{b}}$) is performed using 19.7 fb$^{-1}$ of pp collision data at $\sqrt{s} =$ 8 TeV. The production mechanism investigated in this search is $\mathrm{t\overline{t}}$ pair production in which one top quark decays to a charged Higgs boson and a bottom quark and the other decays to a charged lepton, a neutrino, and a bottom quark. Charged Higgs boson decays to $\mathrm{c\overline{b}}$ are searched for, resulting in a final state containing at least four jets, a charged lepton (muon or electron), and missing transverse momentum. A kinematic fit is performed to identify the pair of jets least likely to be the bottom quarks originating from direct top quark decays and the invariant mass of this pair is used as the final observable in the search. No evidence for the presence of a charged Higgs boson is observed and upper limits at 95% confidence level of 0.8-0.5% are set on the branching fraction $\mathcal{B}$(t $\to$ H$^+$b), assuming $\mathcal{B}$(H$^+\to$ $\mathrm{c\overline{b}}) =$ 1.0 and $\mathcal{B}$(t $\to$ H$^+$b) $+$ $\mathcal{B}$(t $\to$ Wb) = 1.0, for the charged Higgs boson mass range 90-150 GeV.