  GW190521 is a merger of two black holes (BHs), wherein at least one BH lies
within the pair-instability (PI) mass gap, and it is difficult to form because
of the effects of PI supernovae (PISNe) and pulsational PI (PPI). In this
study, we examined the formation of GW190521-like BH-BHs under Population (Pop)
III environments by binary population synthesis calculations. We reveal that
convective overshooting in stellar evolution strongly affects the formation of
GW190521-like BH-BHs. A model with a small overshoot parameter (similar to
GENEC) can form GW190521-like BH-BHs. The derived merger rate is $4 \times
10^{-2}$ ${\rm yr}^{-1}$ ${\rm Gpc}^{-3}$ at a redshift of $\sim 0.82$, which
is comparable to the merger rate of GW190521-like BH-BHs inferred by
gravitational wave (GW) observations. In this model, a $\sim 90~M_\odot$ star
collapses to form a $\sim 90~M_\odot$ BH by avoiding PPI and PISN even if it is
a member of a binary star. This is because it expands up to $10^2~R_\odot$, and
lose only little mass through binary evolution. However, a model with a large
overshoot parameter (similar to Stern) cannot form GW190521-like BH-BHs at all.
Thus, we cannot conclude that a Pop~III binary system is the origin of GW190521
because determination of the overshoot parameter involves highly uncertain. If
a Pop~III binary system is the origin of GW190521, the merger rate of BH-BHs
including a $100-135~M_\odot$ BH is substantially smaller than that of
GW190521-like BH-BHs. This will be assessed by GW observations in the near
future.
