  Since the discovery of GW190521, several proposals have been put forward to
explain the formation of a black hole in the mass gap caused by (pulsational)
pair-instability, $M = 65-130 M_\odot$. We calculate the mass ejection of Pop
III stars by the pulsational pair-instability (PPI) process using a stellar
evolution and hydrodynamical code. If a relatively small, but reasonable value
is adopted for the overshooting parameter, the stars do not become red super
giants during the PPI phase. We show that in this case most of the hydrogen
envelope remains after the mass ejection by PPI. We find that the BH mass could
be at most $M = 109 M_\odot$ below the mass range of pair-instability
supernovae.
