  A measurement of the production cross section of a Higgs boson with
transverse momentum greater than 250 GeV is presented where the Higgs boson
decays to a pair of $\tau$ leptons. It is based on proton-proton collision data
collected by the CMS experiment at the CERN LHC at a center-of-mass energy of
13 TeV. The data sample corresponds to an integrated luminosity of 138
fb$^{-1}$. Because of the large transverse momentum of the Higgs boson the
$\tau$ leptons from its decays are boosted and produced spatially close, with
their decay products overlapping. Therefore, a dedicated algorithm was
developed to reconstruct and identify them. The observed (expected)
significance of the measured signal with respect to the standard model
background-only hypothesis is 3.5 (2.2) standard deviations. The product of the
production cross section and branching fraction is measured to be
1.64$^{+0.68}_{-0.54}$ times the standard model expectation. The fiducial
differential production cross section is also measured as functions of the
Higgs boson and leading jet transverse momenta. This measurement extends the
probed large-transverse-momentum region beyond 600 GeV.
