  Based on $2.7 \times 10^9~\psi(3686)$ decays collected with the BESIII
detector, the radiative decay $\psi(3686)\to\gamma2(\pi^{+}\pi^{-})\eta$ is
investigated to measure properties of S- and P-wave charmonium states. The
branching fraction of the decay $\eta_{c}(1S) \to 2(\pi^{+}\pi^{-})\eta$, which
is found to have a strong dependence on the interference pattern between
$\eta_c(1S)$ and non-$\eta_c(1S)$ processes, is measured in both destructive
and constructive interference scenarios for the first time. The mass and width
of the $\eta_{c}(1S)$ are measured to be $M=(2984.14 \pm 0.13 \pm 0.38)$
MeV/$c^{2}$ and $\Gamma=(28.82 \pm 0.11 \pm 0.82)$ MeV, respectively. Clear
signals for the decays of the $\chi_{cJ}(J=0,1,2)$ and the $\eta_{c}(2S)$ to
$2(\pi^{+}\pi^{-})\eta$ are also observed for the first time, and the
corresponding branching fractions are measured. The ratio of the branching
fractions between the $\eta_{c}(2S)$ and $\eta_{c}(1S)$ decays is significantly
lower than the theoretical prediction, which might suggest different dynamics
in their decays.
