Characteristic and self-centralizing implies coprime automorphism-faithful

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., self-centralizing characteristic subgroup) must also satisfy the second subgroup property (i.e., coprime automorphism-faithful subgroup)
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Statement

Any characteristic Self-centralizing subgroup (?) of a finite group is coprime automorphism-faithful: in particular, it is a Coprime automorphism-faithful characteristic subgroup (?).

Statement with symbols

Let $G$ be a finite group, and $H$ be a characteristic self-centralizing subgroup, i.e. $C_{G}(H)\leq H$ (or equivalently $C_{G}(H)=Z(H)$). Then, any non-identity automorphism of $G$ restricts to a non-identity automorphism of $H$.

Facts used

This is a special case of a somewhat more general fact:

Normal and self-centralizing implies coprime automorphism-faithful