Search for $1^{-+}$ charmonium-like hybrid via $e^{+}{e}^{-}\to \gamma {\eta }^{(\mathrm{\prime })}{\eta }_c$ 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 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 $1^{-+}$ charmonium-like hybrid via $e^{+}{e}^{-}\to \gamma \eta {\eta }_c$ and $e^{+}{e}^{-}\to \gamma {\eta }^{\mathrm{\prime }}{\eta }_c$ 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.