The decays of $\chi_{c2}\to K^{+}K^{-}\pi^{0}$, $K_{S} K^{\pm}\pi^{\mp}$ and $\pi^{+}\pi^{-}\pi^{0}$ are studied with the $\psi(3686)$ data samples collected with the Beijing Spectrometer (BESIII). For the first time, the branching fractions of $\chi_{c2}\to K^{\ast}\overline{K}$, $\chi_{c2}\to a_{2}^{\pm}(1320)\pi^{\mp}/a_{2}^{0}(1320)\pi^{0}$ and $\chi_{c2}\to \rho(770)^{\pm}\pi^{\mp}$ are measured. Here $K^{\ast}\overline{K}$ denotes both $K^{\ast\pm}K^{\mp}$ and $K^{\ast 0}\overline{K}{}^{0}+c.c.$, and $K^{\ast}$ denotes the resonances $K^{\ast}(892)$, $K^{\ast}_{2}(1430)$ and $K^{\ast}_{3}(1780)$. The observations indicate a strong violation of the helicity selection rule in $\chi_{c2}$ decays into vector and pseudoscalar meson pairs. The measured branching fractions of $\chi_{c2}\to K^{\ast}(892)\overline{K}$ are more than 10 times larger than the upper limit of $\chi_{c2}\to \rho(770)^{\pm}\pi^{\mp}$, which is so far the first direct observation of a significant $U$-spin symmetry breaking effect in charmonium decays.