  Using $980~\rm fb^{-1}$ of data on and around the $\Upsilon(nS)(n=1,2,3,4,5)$
resonances collected with the Belle detector at the KEKB asymmetric-energy
$e^+e^-$ collider, the two-photon process $\gamma\gamma\to \gamma\psi(2S)$ is
studied from the threshold to $4.2~{\rm GeV}$ for the first time. Two
structures are seen in the invariant mass distribution of $\gamma\psi(2S)$: one
at $M_{R_1} = 3922.4\pm 6.5 \pm 2.0~{\rm MeV}/c^2$ with a width of
$\Gamma_{R_1} = 22\pm 17\pm 4~{\rm MeV}$, and another at $M_{R_2} = 4014.3\pm
4.0 \pm 1.5~{\rm MeV}/c^2$ with a width of $\Gamma_{R_2} = 4\pm 11 \pm 6~{\rm
MeV}$; the signals are parametrized with the incoherent sum of two Breit-Wigner
functions. The first structure is consistent with the $X(3915)$ or the
$\chi_{c2}(3930)$, and the local statistical significance is determined to be
$3.1\sigma$ with the systematic uncertainties included. The second matches none
of the known charmonium or charmoniumlike states, and its global significance
is determined to be $2.8\sigma$ including the look-elsewhere effect. The
production rates are $\Gamma_{\gamma\gamma}{\cal B}(R_1\to\gamma\psi(2S)) =
9.8\pm 3.6\pm 1.2~{\rm eV}$ assuming $(J^{PC}, |\lambda|) =(0^{++}, 0)$ or
$2.0\pm 0.7\pm 0.2~{\rm eV}$ with $(2^{++}, 2)$ for the first structure and
$\Gamma_{\gamma\gamma}{\cal B}(R_2\to\gamma\psi(2S)) = 6.2\pm 2.2\pm 0.8~{\rm
eV}$ with $(0^{++}, 0)$ or $1.2\pm 0.4\pm 0.2~{\rm eV}$ with $(2^{++}, 2)$ for
the second one. Here, the first errors are statistical and the second
systematic, and $\lambda$ is the helicity.
