Based on $(10087\pm 44)\times {10}^{6}J/\psi$ events collected with the BESIII detector, a partial wave analysis of the decay $J/\psi \to \gamma K_S^0{K}_S^0{\eta }^{\prime }$ is performed. The mass and width of the $X\left(2370\right)$ are measured to be $2395\pm 11\left(\mathrm{stat}{)}_{-94}^{+26}\right(\mathrm{syst})\mathrm{MeV}/c^2$ and ${188}_{-17}^{+18}\left(\mathrm{stat}{)}_{-33}^{+124}\right(\mathrm{syst})\mathrm{MeV}$, respectively. The corresponding product branching fraction is $B[J/\psi \to \gamma X(2370\left)\right]\times B\left[X\right(2370)\to f_0(980\left){\eta }^{\prime }\right]\times B\left[f_0\right(980)\to K_S^0{K}_S^0]=(1.31\pm 0.22(\mathrm{stat}{)}_{-0.84}^{+2.85}\left(\mathrm{syst}\right))\times {10}^{-5}$. The statistical significance of the $X\left(2370\right)$ is greater than $11.7\sigma$ and the spin parity is determined to be $0^{-+}$ for the first time. The measured mass and spin parity of the $X\left(2370\right)$ are consistent with the predictions of the lightest pseudoscalar glueball.