We report on the measurement and investigation of pulsed high-energy $gamma$-ray emission from the Vela pulsar, PSR B0833$-$45, based on 40.3 hours of observations with the largest telescope of H.E.S.S., CT5, in monoscopic mode, and on 8 years of data obtained with the Fermi-LAT. A dedicated very-low-threshold event reconstruction and analysis pipeline was developed and, together with the CT5 telescope response model, was validated using the Fermi-LAT data as reference. A pulsed $gamma$-ray signal at a significance level of more than $15sigma$ is detected from the P2 peak of the Vela pulsar light curve. Of a total of 15835 events, more than 6000 lie at an energy below 20 GeV, implying a significant overlap between H.E.S.S. II-CT5 and the Fermi-LAT. While the investigation of the pulsar light curve with the LAT confirms characteristics previously known up to 20 GeV, in the tens of GeV energy range, CT5 data show a change in the pulse morphology of P2, i.e., an extreme sharpening of its trailing edge, together with the possible onset of a new component at 3.4$sigma$ significance level. Assuming a power-law model for the P2 spectrum, an excellent agreement is found for the photon indices ($Gamma simeq$ 4.1) obtained with the two telescopes above 10 GeV and an upper bound of 8% is derived on the relative offset between their energy scales. Using both instruments data, it is however shown that the spectrum of P2 in the 10-100 GeV has a pronounced curvature, i.e. a confirmation of the sub-exponential cutoff form found at lower energies with the LAT. This is further supported by the weak evidence for an emission above 100 GeV obtained with CT5. In contrast, converging indications are found from both CT5 and LAT data for the emergence of a hard component above 50 GeV in the leading wing (LW2) of P2, which possibly extends beyond 100 GeV.
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