Characteristic and self-centralizing implies coprime automorphism-faithful

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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) \le 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