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This paper presents the third data release of the INvestigating StellarPopulation In RElics (INSPIRE) project, comprising 52 ultra-compact massivegalaxies (UCMGs) observed with the X-Shooter spectrograph. We measureintegrated stellar velocity dispersion, [Mg/Fe] abundances, ages, andmetallicities for all the INSPIRE objects. We thus infer star formationhistories and confirm the existence of a degree of relicness (DoR), defined interms of the fraction of stellar mass formed by $z=2$, the time at which agalaxy has assembled 75\% of its mass, and the final assembly time. Objectswith a high DoR assembled their stellar mass at early epochs, while low-DoRobjects show a non-negligible fraction of later-formed populations and hence aspread in ages and metallicities. A higher DoR correlates with larger [Mg/Fe],super-solar metallicity, and larger velocity dispersion values. The 52 UMCGsspan a large range of DoR from 0.83 to 0.06, with 38 of them having formed morethan 75\% of their mass by $z=2$. Of these, nine are extreme relics(DoR$>0.7$), since they formed the totality ($>99\%$) of their stellar mass byredshift $z=2$. The remaining 14 UCMGs cannot be considered relics, as they arecharacterised by more extended star formation histories. With INSPIRE, we builtthe first sizeable sample of relics outside the local Universe, up to$z\sim0.4$, increasing the number of confirmed relics by a factor of $>10$, andopening up an important window to explain the mass assembly of massive galaxiesin the high-z Universe.

INSPIRE: INvestigating Stellar Population In RElics V. A catalogue of  ultra-compact massive galaxies outside the local Universe and their degree of  relicness