Stabilizer is Normal iff Stabilizer of Each Element of Orbit

Theorem
Let $\struct {G, \circ}$ be a group.

Let $S$ be a set.

Let $*: G \times S \to S$ be a group action.

Let $x \in S$.

Let $\Stab x$ denote the stabilizer of $x$ under $*$.

Let $\Orb x$ denote the orbit of $x$ under $*$.

Then $\Stab x$ is normal in $G$ $\Stab x$ is also the stabilizer of every element in $\Orb x$.

Necessary Condition
Let $\Stab x$ be normal in $G$.

$y \in \Orb x$ such that $\Stab x \ne \Stab y$.

Then:

Sufficient Condition
Let $\Stab x$ be the stabilizer of every element in $\Orb x$.