Condition for Group Endomorphism to Commute with All Inner Automorphisms

Theorem
Let $G$ be a group.

Let $\phi: G \to G$ be an endomorphism on $G$.

Let $\phi$ be such that:
 * $\forall a \in G: \kappa_a \circ h = h \circ \kappa_a$

where:
 * $\kappa_a$ denotes the inner automorphism of $G$ given by $a$
 * $\circ$ denotes composition of mappings.

Then:
 * $H = \set {x \in G: \map \phi {\map \phi x} = \map \phi x}$

is a normal subgroup of $G$

Also, the quotient group $G / H$ is an abelian group.