Search for $ {1}^{-+}$ charmoniumlike hybrid via $ {e}^{+}{e}^{-}\to \gamma {\eta }^{(\prime )}{\eta }_{c}$ at center-of-mass energies between 4.258 and 4.681 GeV

Using $ {e}^{+}{e}^{-}$ collision data corresponding to an integrated luminosity of $ 10.6\text{}\text{}{\mathrm{fb}}^{-1}$ collected at center-of-mass energies between 4.258 and 4.681 GeV with the BESIII detector at the BEPCII collider, we search for the $ {1}^{-+}$ charmoniumlike hybrid via $ {e}^{+}{e}^{-}\to \gamma \eta {\eta }_{c}$ and $ {e}^{+}{e}^{-}\to \gamma {\eta }^{\prime }{\eta }_{c}$ decays for the first time. No significant signal is observed, and the upper limits on the Born cross sections for both processes are set at the 90% confidence level.