  We calculate new evolutionary models of rotating primordial very massive
stars, with initial mass from $100\,M_{\odot}$ to $200\,M_{\odot}$, 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 ${\sim20}$ using current telescopes such as the Swift-BAT X-ray
detector and the JWST.
