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Motivated by the first untagged decay-time-integrated amplitude analysis of$B_s \to K_SK^{\mp}\pi^{\pm}$ decays performed by LHCb collaboration, where thedecay amplitudes are modeled to contain the resonant contributions fromintermediate resonances $K^*(892)$, $K_0^*(1430)$ and $K_2^*(1430)$, wecomprehensively investigate the quasi-two-body $B_{s} \to K^0/\overline{\kern-0.2em K}^0 K^{\pm}\pi^{\mp}$ decays, and calculate the branching fractions andthe time-dependent $CP$ asymmetries within the perturbative QCD approach basedon the $k_T$ factorization. In the quasi-two-body space region the calculatedbranching fractions with the considered intermediate resonances are in goodagreement with the experimental results of LHCb by adopting proper $K\pi$ pairwave function, describing the interaction between the kaon and pion in the$K\pi$ pair. Furthermore,within the obtained branching fractions of thequasi-two-body decays, we also calculate the branching fractions ofcorresponding two-body decays, and the results consist with the LHCbmeasurements and the earlier studies with errors. For these considered decays,since the final states are not flavour-specific, the time-dependent $CP$ couldbe measured. We calculate six $CP$-violation observables, which can be testedin the ongoing LHCb experiment.

Branching Fractions and CP Asymmetries of the Quasi-Two-Body Decays in $B_{s} \to K^0(\overline K^0)K^\pm \pi^\mp$ within PQCD Approach