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Here, we present measurements of turbulent drag reduction in boundary layers at high friction Reynolds numbers in the range of $4500 \le Re_τ\le 15000$. The efficacy of the approach, using streamwise travelling waves of spanwise wall oscillations, is studied for two actuation regimes: (i) inner-scaled actuation (ISA), as investigated in Part 1 of this study, which targets the relatively high-frequency structures of the near-wall cycle, and (ii) outer-scaled actuation (OSA), which was recently presented by Marusic et al. (Nat. Commun., vol. 12, 2021) for high-$Re_τ$ flows, targeting the lower-frequency, outer-scale motions. Multiple experimental techniques were used, including a floating-element balance to directly measure the skin-friction drag force, hot-wire anemometry to acquire long-time fluctuating velocity and wall-shear stress, and stereoscopic-PIV (particle image velocimetry) to measure the turbulence statistics of all three velocity components across the boundary layer. Under the ISA pathway, drag reduction of up to 25% was achieved, but mostly with net power saving losses due to the high-input power cost associated with the high-frequency actuation. The low-frequency OSA pathway, however, with its lower input power requirements, was found to consistently result in positive net power savings of 5 - 10%, for moderate drag reductions of 5 - 15%. The results suggest that OSA is an attractive pathway for energy-efficient drag reduction in high Reynolds number applications. Both ISA and OSA strategies are found to produce complex inter-scale interactions, leading to attenuation of the turbulent fluctuations across the boundary layer for a broad range of length and time scales.