  This is the second data release (DR2) of the INvestigating Stellar Population
In RElics (INSPIRE) project, comprising 21 new systems with observations
completed before March 2022. For each system, we release four one-dimensional
(1D) spectra to the ESO Science Archive, one spectrum for each arm of the
X-Shooter spectrograph. In this paper, we focus on the line-of-sight velocity
distribution, measuring integrated stellar velocity dispersions from the
spectra, and assessing their robustness and the associated uncertainties. For
each of the 21 new systems, we systematically investigated the effect of the
parameters and set-ups of the full spectral fitting on the stellar velocity
dispersion ($\sigma$) measurements. In particular, we tested how $\sigma$
changes when several parameters of the fit as well as the resolution and
spectral coverage of the input spectra are varied. We found that the effect
that causes the largest systematic uncertainties on $\sigma$ is the wavelength
range used for the fit, especially for spectra with a lower signal-to-noise
ratio (S/N $\leq$ 30). When using blue wavelengths (UVB arm) one generally
underestimates the velocity dispersion (by $\sim$15 km/s). The values obtained
from the near-IR (NIR) arm present a larger scatter because the quality of the
spectra is lower. We finally compared our results with those in literature,
finding a very good agreement overall. Joining results obtained in DR1 with
those presented here, INSPIRE contains 40 ultra-compact massive galaxies,
corresponding to 75% of the whole survey. By plotting these systems in a
stellar mass-velocity dispersion diagram, we identify at least four highly
reliable relic candidates among the new systems. Their velocity dispersion is
larger than that of normal-sized galaxies of similar stellar mass.
