Characteristic and self-centralizing implies coprime automorphism-faithful

Statement
Any characteristic fact about::self-centralizing subgroup of a finite group is coprime automorphism-faithful: in particular, it is a fact about::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