Let $G$ be a group, $H$ a subgroup of $G$ and $N$ the normalizer of $H$. Then there exist a homomorphism of the opposite quotient group $(N/H)^{op}$ into the symmetric group of the quotient set $G/H$: $$\phi:(N/H)^{op}\rightarrow\mathfrak{S}_{G/H}.$$ Moreover $\phi$ induces an isomorphism of $(N/H)^{op}$ onto the group of automorphisms of the $G$-set $G/H$.
Suppose $G$ is a group. Then $G$ operates on itself by left translation: $G$ is a left homogeneous principal $G$-set; denote it by $G_s$ .
How does the quoted result imply that the group of automorphisms of the homogeneous principal $G$-set $G_s$ is isomorphic to $G^{op}$?