  In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with
dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries
on $M$ with cohomogeneity at least $3$. Suppose the union of non-principal
orbits $M\setminus M^{reg}$ is a smooth embedded submanifold of $M$ without
boundary and ${\rm dim}(M\setminus M^{reg})\leq n-2 $. Then for any
$c\in\mathbb{R}$, we show the existence of a nontrivial, smooth, closed,
$G$-equivariant almost embedded $G$-invariant hypersurface $\Sigma^n$ of
constant mean curvature $c$.
