  An analysis of the $J/\psi\to \eta\eta^\prime\gamma$ decay by the BESIII
collaboration claims the observation of an exotic state $\eta_1(1855)$ with
$I^GJ^{PC}=0^+1^{-+}$. To establish its C-parity partner
$\eta^\prime_{1}(1855)$ in the picture of the $K \bar{K}_1(1400)$ molecular
state, we propose that $J/\psi\to\eta^\prime_{1}(1855)\eta^{(\prime)}$ receives
the main contributions from the final state interaction of $KK^*$($K^+ K^{*-}$,
$K^- K^{*+}$, $K^0 \bar{K}^{*0}$, and $\bar{K}^0 K^{*0}$). Specifically, $K$
and $K^*$ in $J/\psi\to KK^*$ decays transform as
$\eta^\prime_{1}(1855)\eta^{(\prime)}$, by exchanging a $K_1(1400)$. We predict
${\cal B}(J/\psi\to \eta^{\prime}_1(1855)\eta)=(6.3^{+12.6}_{-3.5})\times
10^{-6}$, and ${\cal B}(J/\psi\to \eta^{\prime}_1(1855)\eta^{\prime})
=(6.5^{+6.6}_{-4.6})\times 10^{-6}$, which can be studied in the $J/\psi\to
K^*\bar{K}^*\eta^{(\prime)}$ decays.
