  Using a sample of $4.48\times10^{8}$ $\psi(3686)$ events collected with the
BESIII detector at the BEPCII collider, we study the two-photon decays of the
pseudoscalar mesons $\pi^0$, $\eta$, $\eta^\prime$, $\eta(1405)$, $\eta(1475)$,
$\eta(1760)$, and $X(1835)$ in $J/\psi$ radiative decays using
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$ events. The $\pi^0$, $\eta$ and
$\eta^\prime$ mesons are clearly observed in the two-photon mass spectra, and
the branching fractions are determined to be
$B(J/\psi\to\gamma\pi^{0}\to3\gamma)=(3.57\pm0.12\pm0.16)\times10^{-5}$,
$B(J/\psi\to\gamma\eta\to3\gamma)=(4.42\pm0.04\pm0.18)\times10^{-4}$, and
$B(J/\psi\to\gamma\eta'\to3\gamma)=(1.26\pm0.02\pm0.05)\times10^{-4}$, where
the first errors are statistical and the second systematic. No clear signal for
$\eta(1405)$, $\eta(1475)$, $\eta(1760)$ or $X(1835)$ is observed in the
two-photon mass spectra, and upper limits at the $90\%$ confidence level on the
product branching fractions are obtained.
