The primary objective of this study is to investigate hadronic molecules of $K^{⁎}{\overline{K}}_1\left(1270\right)$ using a one-boson-exchange model, which incorporates exchanges of vector and pseudoscalar mesons in the t-channel, as well as the pion exchange in the u-channel. Additionally, careful consideration is given to the three-body effects resulting from the on-shell pion originating from $K_1\left(1270\right)\to K^{⁎}\pi$. Then the BESIII data of the $J/\psi \to \varphi \eta {\eta }^{\prime }$ process is fitted using the $K^{⁎}{\overline{K}}_1\left(1270\right)$ scattering amplitude with $J^{PC}=0^{--}$ or $1^{--}$. The analysis reveals that both the $J^{PC}=0^{--}$ and $1^{--}$ assumptions for $K^{⁎}{\overline{K}}_1\left(1270\right)$ scattering provide good descriptions of the data, with similar fit qualities. Notably, the parameters obtained from the best fits indicate the existence of $K^{⁎}{\overline{K}}_1\left(1270\right)$ bound states, denoted by $\varphi \left(2100\right)$ and ${\varphi }_0\left(2100\right)$ for the $1^{--}$ and $0^{--}$ states, respectively. The current experimental data, including the η polar angular distribution, cannot distinguish which $K^{⁎}{\overline{K}}_1\left(1270\right)$ bound state contributes to the $J/\psi \to \varphi \eta {\eta }^{\prime }$ process, or if both are involved. Therefore, we propose further explorations of this process, as well as other processes, in upcoming experiments with many more $J/\psi$ events to disentangle the different possibilities.