Measurements of the branching fractions of ${\chi }_{cJ}\to \phi \phi \eta ,\phi \phi {\eta }^{\mathrm{\prime }}$ and $\phi K^{+}{K}^{-}\eta$

Using a sample of $(2712.4\pm 14.3)\times {10}^6\psi$(3686) events collected by the BESIII detector at the BEPCII collider, we measure the branching fractions of the decays ${\chi }_{cJ}\to \phi \phi \eta ,\phi \phi {\eta }^{\mathrm{\prime }}$, and~$\phi K^{+}{K}^{-}\eta$ ($J=0,1,2$). The obtained branching fractions are $\mathcal{B}({\chi }_{c0}\to \phi \phi \eta )=(7.40\pm 0.23\pm 0.55)\times {10}^{-4}$, $\mathcal{B}({\chi }_{c1}\to \phi \phi \eta )=(3.33\pm 0.14\pm 0.25)\times {10}^{-4}$, $\mathcal{B}({\chi }_{c2}\to \phi \phi \eta )=(5.46\pm 0.17\pm 0.40)\times {10}^{-4}$, $\mathcal{B}({\chi }_{c0}\to \phi \phi {\eta }^{\mathrm{\prime }})=(2.96\pm 0.23\pm 0.29)\times {10}^{-4}$, $\mathcal{B}({\chi }_{c1}\to \phi \phi {\eta }^{\mathrm{\prime }})=(0.69\pm 0.10\pm 0.08)\times {10}^{-4}$, $\mathcal{B}({\chi }_{c2}\to \phi \phi {\eta }^{\mathrm{\prime }})=(0.65\pm 0.09\pm 0.07)\times {10}^{-4}$, $\mathcal{B}({\chi }_{c0}\to \phi K^{+}{K}^{-}\eta )=(1.23\pm 0.08\pm 0.10)\times {10}^{-4}$, $\mathcal{B}({\chi }_{c1}\to \phi K^{+}{K}^{-}\eta )=(1.00\pm 0.07\pm 0.07)\times {10}^{-4}$, and $\mathcal{B}({\chi }_{c2}\to \phi K^{+}{K}^{-}\eta )=(1.82\pm 0.09\pm 0.14)\times {10}^{-4}$, where $K^{+}{K}^{-}$ is not from the decay of a $\phi$ meson, the first uncertainties are statistical and the second systematic. The branching fractions of ${\chi }_{cJ}\to \phi \phi \eta$ are measured with precision improved by factors of $1.5-1.9$, and those of ${\chi }_{cJ}\to \phi \phi {\eta }^{\mathrm{\prime }}$ and $\phi K^{+}{K}^{-}\eta$ are measured for the first time.