Characteristic implies automorph-conjugate

Verbal statement
Any characteristic subgroup of a group is an automorph-conjugate subgroup.

Symbolic statement
Let $$H$$ be a characteristic subgroup of $$G$$. Then $$H$$ is an automorph-conjugate subgroup of $$G$$.

Property-theoretic statement
The subgroup property of being characteristic is stronger than the subgroup property of being automorph-connjugate.

Characteristic subgroup
A subgroup of a group is termed a characteristic subgroup if any automorphism of the group maps the subgroup to itself.

That is, $$H \le G$$ is characteristic if for any automorphism $$\sigma$$ of $$G$$, $$\sigma(H)=H$$.

Characteristicity can also be expressed using the relation implication expression:

Automorphic subgroups $$\implies$$ Equal subgroups

In other words, any subgroup obtained as $$\sigma(H)$$ where $$\sigma$$ is an automorphism of $$G$$, is equal to $$H$$.

Automorph-conjugate subgroup
A subgroup of a group is termed an automorph-conjugate subgroup if any automorphism of the group maps the subgroup to a conjugate subgroup.

That is $$H \le G$$ is automorph-conjugate if for any automorphism $$\sigma$$ of $$G$$ there exists $$g \in G$$ such that $$\sigma(H) = gHg^{-1}$$ (the latter is sometimes denoted $$H^g$$).

This property can also be expressed using the relation implication expression:

Automorphic subgroups $$\implies$$ Conjugate subgroups

In other words, any subgroup obtained as $$\sigma(H)$$ where $$\sigma$$ is an automorphism of $$G$$, is also a conjugate subgroup to $$H$$.

Hands-on proof
Given: $$H$$ is a characteristic subgroup of $$G$$,

To prove: $$H$$ is an automorph-conjugate subgroup of $$G$$. In other words, for any automorphism $$\sigma$$ of $$G$$, there exists $$g \in G$$, $$\sigma(H) = H^g$$.

Proof: For any automorphism $$\sigma$$ of $$G$$, $$\sigma(H) = H$$ (by definition of characteristicity). Clearly $$H$$ is a conjugate subgroup to itself (say $$H^e = H$$ where $$e$$ is the identity element). Thus, $$H$$ is automorph-conjugate.

The property of being a characteristic subgroup can be expressed as a relation implication:

Automorphic subgroups $$\implies$$ Equal subgroups

The property of being an automorph-conjugate subgroup can be expressed as a relation implication:

Automorphic subgroups $$\implies$$ Conjugate subgroups

Since the left side is the same for both properties, but the right side is stronger for characteristicity, we see that characteristic implies automorph-conjugate.

Converse
The converse is not true. Counterexamples can be easily constructed by looking at Sylow subgroups which are not normal. This is because any Sylow subgroup must be an isomorph-conjugate subgroup, and hence an automorph-conjugate subgroup.