  The current gravitational wave (GW) detectors have successfully observed many
binary compact objects, and the third generation ground-based GW detectors such
as Einstein telescope and space-borne detectors such as LISA will start their
GW observation in a decade. Ahead of the arrival of this new era, we perform a
binary population synthesis calculation for very massive ($\sim$ 100--1000
$M_\odot$) population (Pop.) III stars, derive the various property of binary
black hole (BBH) mergers with intermediate mass black holes (IMBHs) and
investigate the dependence on common envelope parameter $\alpha\lambda$ which
is still not a well understood parameter. We find that the maximum mass of
primary BH mass is larger for smaller value of common envelope parameter. In
this study, we adopt double power law initial mass function (IMF) for Pop. III
stars, and put some constraints on Pop. III IMF by comparing our obtained
merger rate density at the local Universe with that derived from gravitational
wave (GW) observation. We compute the detection rate and show that the third
generation ground-based GW detector, Einstein telescope, have a potential to
detect $\sim$ 10--1000 BBHs with IMBHs per year. We also find that we may be
able to obtain the insight into $\alpha\lambda$ if a BBH with total mass
$\gtrsim500M_\odot$ are detected by advanced LIGO (O4) or LISA.
