  We search for the hadronic decay $\eta_c(2S)\to 2(\pi^+\pi^-)$ in the
$\psi(3686)\to\gamma\eta_c(2S)$ radiative decay using $(27.12\pm 0.14)\times
10^8$ $\psi(3686)$ events collected by the BESIII detector at the BEPCII
collider. No significant signal is found, and the upper limit of
$\mathcal{B}[\psi(3686)\to\gamma\eta_c(2S)]\mathcal{B}[\eta_c(2S)\to
2(\pi^+\pi^-)]$ is determined to be $0.78\times 10^{-6}$ at the 90\% confidence
level. Using $\psi(3686)\to\gamma\chi_{cJ}$ transitions, we also measure the
branching fractions of $\mathcal{B}[\chi_{cJ(J=0,1,2)}\to 2(\pi^+\pi^-)]$,
which are $\mathcal{B}[\chi_{c0}\to 2(\pi^+\pi^-)]=(2.127\pm
0.002~(\mathrm{stat.})\pm 0.101~(\mathrm{syst.}))$\%, $\mathcal{B}[\chi_{c1}\to
2(\pi^+\pi^-)]=(0.685\pm 0.001~(\mathrm{stat.})\pm 0.031~\mathrm{syst.}))$\%,
and $\mathcal{B}[\chi_{c2}\to 2(\pi^+\pi^-)]=(1.153\pm
0.001~(\mathrm{stat.})\pm 0.063~(\mathrm{syst.}))$\%.
