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Since the discovery of GW190521, several proposals have been put forward toexplain 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 PopIII stars by the pulsational pair-instability (PPI) process using a stellarevolution and hydrodynamical code. If a relatively small, but reasonable valueis adopted for the overshooting parameter, the stars do not become red supergiants during the PPI phase. We show that in this case most of the hydrogenenvelope remains after the mass ejection by PPI. We find that the BH mass couldbe at most $M = 109 M_\odot$ below the mass range of pair-instabilitysupernovae.

Pulsational pair-instability and the mass gap of Population III Black Holes: Effects of overshooting