  Using a data sample of 106 million $\psi(3686)$ events collected with the
BESIII detector operated at the BEPCII storage ring, we study for the first
time the decay $\chi_{cJ}\to\phi K^{0}_S K^{\pm}\pi^{\mp}$ and
$\chi_{cJ}\to\phi K^{+} K^{-}\pi^{0}$ in the E1 radiative transition
$\psi(3686)\to\gamma\chi_{cJ}$. The decays are dominated by the three-body
decay $\chi_{cJ}\to \phi K^*(892)\bar{K}$. We measure branching fractions for
this reaction via the neutral and charged $K^*(892)$ and find them consistent
with each other within the expectation of isospin symmetry. In the
$K\bar{K}\pi$ invariant mass distribution a structure near the
$K^*(892)\bar{K}$ mass threshold is observed, and the corresponding mass and
width are measured to be $1412\pm4(\mathrm{stat.})\pm8(\mathrm{sys.})
\mathrm{MeV}/c^2$ and $\Gamma$ = $84\pm12(\mathrm{stat.})\pm40(\mathrm{sys.})
\mathrm{MeV}$, respectively. The observed state favors an assignment to the
$h_1(1380)$, considering its possible $J^{PC}$ and comparing its mass, width
and decay mode to those reported in the Particle Data Group.
