Fully invariant implies characteristic

Statement
Any fully invariant subgroup (also called fully characteristic subgroup) of a group is a characteristic subgroup.

Fully invariant subgroup
A subgroup $$H$$ of a group $$G$$ is termed fully invariant in $$G$$ if, for every endomorphism $$\varphi$$ of $$G$$, $$\varphi(H)$$ is contained in $$H$$.

Characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed characteristic in $$G$$ if, for every automorphism $$\varphi$$ of $$G$$, $$\varphi(H)$$ is contained in $$H$$.

Related facts

 * Characteristic not implies fully invariant
 * Characteristic not implies fully invariant in finite abelian group
 * Characteristic equals fully invariant in odd-order abelian group

Invariance under restricted classes of endomorphisms

 * Strictly characteristic subgroup is a subgroup that is invariant under all surjective endomorphisms. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.
 * Injective endomorphism-invariant subgroup is a subgroup that is invariant under all injective endomorphisms. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.

Proof idea
The idea behind the proof is that since every automorphism is an endomorphism, invariance under all endomorphisms implies invariance under automorphisms.

The property of being fully invariant is the invariance property with respect to endomorphisms. It has the function restriction expression:

Endomorphism $$\to$$ Function

The property of being characteristic is the invariance property with respect to automorphisms. It has the function restriction expression:

Automorphism $$\to$$ Function

Since the left side of the function restriction expression for characteristicity is stronger while the right sides of both function restriction expressions are equal, the property of being characteristic is weaker than the property of being fully invariant.