We calculate new evolutionary models of rotating primordial very massive stars, with initial mass from 100 M circle dot to 200 M circle dot, for two values of the initial metallicity Z = 0 and Z = 0.0002. For the first time in this mass range, we consider stellar rotation and pulsation-driven mass loss, along with radiative winds. The models evolve from the zero-age main sequence until the onset of pair-instability. We discuss the main properties of the models during their evolution and then focus on the final fate and the possible progenitors of jet-driven events. All tracks that undergo pulsational-pair instability produce successful gamma-ray bursts (GRB) in the collapsar framework, while those that collapse directly to black holes (BH) produce jet-driven supernova events. In these latter cases, the expected black hole mass changes due to the jet propagation inside the progenitor, resulting in different models that should produce BH within the pair-instability black hole mass gap. Successful GRBs predicted here from zero metallicity, and very metal-poor progenitors, may be bright enough to be detected even up to redshift similar to 20 using current telescopes such as the Swift-BAT X-ray detector and the JWST.
