  Let $G$ be a compact connected subgroup of $SO(n+1)$. In $\mathbb{R}^{n+1}$,
we gain interior $G$-symmetry for minimal hypersurfaces and hypersurfaces of
constant mean curvature (CMC) which have $G$-invariant boundaries and
$G$-invariant contact angles along boundaries. The main ingredients of the
proof are to build an associated Cauchy problem based on infinitesimal Lie
group actions, and to apply Morrey's regularity theory and the
Cauchy-Kovalevskaya Theorem. Moreover, we also investigate the same kind of
symmetry inheritance from boundaries for hypersurfaces of constant higher order
mean curvature and Helfrich-type hypersurfaces in $\mathbb{R}^{n+1}$.
