Inner Automorphism Maps Subgroup to Itself iff Normal

Theorem
Let $G$ be a group.

For $x \in G$, let $\kappa_x$ denote the inner automorphism of $x$ in $G$.

Let $H$ be a subgroup of $G$.

Then:
 * $\forall x \in G: \kappa_x \sqbrk H = H$


 * $H$ is a normal subgroup of $G$.
 * $H$ is a normal subgroup of $G$.