We obtain the interior symmetry of embedded $C^1$ minimal hypersurfaces and hypersurfaces of constant mean curvature in $\mathbb{R}^{n+1}$ with $G$-invariant boundary and $G$-invariant contact angle, where $G$ is a compact connected Lie subgroup of $SO(n+1)$. This extends the result for spherical boundaries where $G=SO(n)$. The main idea is to build a real analytic solution of a Cauchy problem based on infinitesimal Lie group actions and Morrey's regularity theory. It allows us to apply Cauchy-Kovalevskaya theorem. By the same argument, we also investigate the symmetry inheritance from boundaries for hypersurfaces of constant higher order mean curvature and Helfrich-type hypersurfaces in $\mathbb{R}^{n+1}$. This talk is based on the recent joint work with Chao Qian, Jing Wu and Yongsheng Zhang.