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The primary objective of this study is to investigate hadronic molecules of K ⁎ K ¯ 1 (1270) 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 (1270) → K ⁎ π. Then the BESIII data of the J / ψ → ϕ η η ′ process is fitted using the K ⁎ K ¯ 1 (1270) scattering amplitude with J P C = 0 − − or 1 − −. The analysis reveals that both the J P C = 0 − − and 1 − − assumptions for K ⁎ K ¯ 1 (1270) scattering provide good descriptions of the data, with similar fit qualities. Notably, the parameters obtained from the best fits indicate the existence of K ⁎ K ¯ 1 (1270) bound states, denoted by ϕ (2100) and ϕ 0 (2100) for the 1 − − and 0 − − states, respectively. The current experimental data, including the η polar angular distribution, cannot distinguish which K ⁎ K ¯ 1 (1270) bound state contributes to the J / ψ → ϕ η η ′ 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 / ψ events to disentangle the different possibilities. [ABSTRACT FROM AUTHOR]