Foliation of a homogeneous manifold and invarience of leaves

Question

Let $M=G/H$ be a foliated homogeneous manifold, where $G$ is a connected Lie group. Let $L$ be a leaf of this foliation. Suppose that $x\in L$ and $k\in G$ be an element such that $k\cdot x\in L$. How to prove that $gkg^{-1}\cdot x\in L$ for any $g\in G$?