On coset-spaces of compact Lie groups by subgrops of corank two

Let $G$ and $G'$ be compact connected simply connected Lie groups. Suppose that $G$ is a simple group, $L$ is a centralizer of torus of $G$, $H$ is the commutant of the subgroup $L$, and $\mathrm{rk}~G - \mathrm{rk}~H = 2$. Let $\mathrm{Sam}~(G/H) = 0$. We prove the following assertion: if the group $G'$ acts transitively and almost effectively on the manifold $G/H$, then $G' \cong G$. If all lenghts of roots of the group $G$ are equal, then the action of $G'$ on $G/H$ is similar to the action of $G$.