  We study the production of the light $J^{PC}=1^{-+}$ hybrid isoscalars
$\eta_1^\prime(1855)$ and the yet undiscovered $\eta_1^{hyb}(1660)$ as
intermediate states in the radiative decays of the charmonium ($J/\psi$) to two
conventional mesons using a flavor symmetric Lagrangian. For this purpose, we
use the $J/\psi\to\gamma\eta_1^\prime(1855)\to\gamma\eta\eta^\prime$ process as
the reference. We find that some of the decay channels have branching ratios
similar to or larger than that of the $\gamma\eta\eta^\prime$ channel and are
sensitive to the mixing between the hybrid isoscalars. We propose that
relatively stable $\gamma\eta f_1(1285)$ channel be explored for the presence
of the light hybrid isoscalar $\eta_1^{hyb}(1660)$. We also exploit the strong
decay channels containing at least one vector meson to study the radiative
decays of the whole hybrid nonet
$\{\pi_1(1600),K_1^{hyb}(1750),\eta_1^{hyb}(1660),\eta_1'(1855)\}$. We find
that the hybrids cannot radiatively decay into the $I=0$ pseudoscalars.
Furthermore, the vector decay channels ($(\rho/\omega/\phi)\gamma$) of the
hybrid isoscalars are sensitive to the strangeness content of the hybrids. We
also provide estimates for the branching fractions for the radiative production
and partial widths for the radiative decays of the hybrids.
