  We perform a ground-based near-infrared spectroscopic survey using the
Keck/MOSFIRE spectrograph to target Ly$\alpha$ emission at $7.0<z<8.2$ from 61
galaxies to trace the ionization state of the intergalactic medium (IGM). We
cover a total effective sky area of $\sim10^\prime\times10^\prime$ in the
Extended Groth Strip field of the Cosmic Assembly Near-infrared Deep
Extragalactic Legacy Survey. From our observations, we detect Ly$\alpha$
emission at a $>$4$\sigma$ level in eight $z>7$ galaxies, which include
additional members of the known $z\sim7.7$ Ly$\alpha$-emitter (LAE) cluster
(Tilvi et al. 2020). With the addition of these newly-discovered $z\sim7.7$
LAEs, this is currently the largest measured LAE cluster at $z>7$. The
unusually-high Ly$\alpha$ detection rate at $z\sim7.7$ in this field suggests
significantly stronger Ly$\alpha$ emission from the clustered LAEs than from
the rest of our targets. We estimate the ionized bubble sizes around these LAEs
and conclude that the LAEs are clustered within an extended ionized structure
created by overlapping ionized bubbles which allow the easier escape of
Ly$\alpha$ from galaxies. It is remarkable that the brightest object in the
cluster has the lowest measured redshift of the Ly$\alpha$ line, being placed
in front of the other LAEs in the line-of-sight direction. This suggests that
we are witnessing the enhanced IGM transmission of Ly$\alpha$ from galaxies on
the rear side of an ionized area. This could be a consequence of Ly$\alpha$
radiative transfer: Ly$\alpha$ close to the central velocity is substantially
scattered by the IGM while Ly$\alpha$ from the rear-side galaxies is
significantly redshifted to where it has a clear path.
