  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 the
decay amplitudes are modeled to contain the resonant contributions from
intermediate resonances $K^*(892)$, $K_0^*(1430)$ and $K_2^*(1430)$, we
comprehensively 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 and
the time-dependent $CP$ asymmetries within the perturbative QCD approach based
on the $k_T$ factorization. In the quasi-two-body space region the calculated
branching fractions with the considered intermediate resonances are in good
agreement with the experimental results of LHCb by adopting proper $K\pi$ pair
wave function, describing the interaction between the kaon and pion in the
$K\pi$ pair. Furthermore,within the obtained branching fractions of the
quasi-two-body decays, we also calculate the branching fractions of
corresponding two-body decays, and the results consist with the LHCb
measurements and the earlier studies with errors. For these considered decays,
since the final states are not flavour-specific, the time-dependent $CP$ could
be measured. We calculate six $CP$-violation observables, which can be tested
in the ongoing LHCb experiment.
