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A search for a heavy Higgs boson in the mass range from 0.2 to 3.0 TeV, decaying to a pair of W bosons, is presented. The analysis is based on proton-proton collisions at $\sqrt{s} =$ 13 TeV recorded by the CMS experiment at the LHC in 2016, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The W boson pair decays are reconstructed in the 2$\ell$2$\nu$ and $\ell\nu$2q final states (with $\ell =$ e or $\mu$). Both gluon fusion and vector boson fusion production of the signal are considered. Interference effects between the signal and background are also taken into account. The observed data are consistent with the standard model (SM) expectation. Combined upper limits at 95% confidence level on the product of the cross section and branching fraction exclude a heavy Higgs boson with SM-like couplings and decays up to 1870 GeV. Exclusion limits are also set in the context of a number of two-Higgs-doublet model formulations, further reducing the allowed parameter space for SM extensions.