  We present an exploratory $N_f=2$ lattice QCD study of $\psi_2(3823)\to
\gamma \chi_{c1}$ at a pion mass $m_{\pi}\approx 350$~MeV. The related
two-point and three-piont functions are calculated using the distillation
method. The electromagnetic multipole form factor $\hat{V}(0)=2.083(11)$ for
$J/\psi\to\gamma \eta_c$ is consistent with previous lattice results, the form
factors $\hat{E}_1(0)$, $\hat{M}_2(0)$ and $\hat{E}_3(0)$ for
$\Gamma(\chi_{c2}\to\gamma J/\psi)$ have the same hierarchy as that derived
from experiments and the predicted decay width $\Gamma(\chi_{c2}\to\gamma
J/\psi)=368(5)~\text{keV}$ is in excellent agreement with the PDG value
$374(10)~\text{keV}$ and previous lattice QCD results in the quenched
approximation. The same strategy is applied to the study of the process
$\psi_2(3823)\to \gamma \chi_{c1}$ and the partial decay width is predicted to
be $337(27)~\text{keV}$. According to the BESIII constraints on the
$\psi_2(3823)$ decay channels and some phenomenological results, we estimate
the total width $\Gamma(\psi_2(3823))=520(100)~\text{keV}$.
