Coprime automorphism-faithful characteristicity is transitive

Property-theoretic statement
The subgroup property of being a coprime automorphism-faithful characteristic subgroup satisfies the subgroup metaproperty of being transitive.

Statement with symbols
Suppose $$G$$ is a finite group, $$K$$ is a coprime automorphism-faithful characteristic subgroup of $$G$$, and $$H$$ is a coprime automorphism-faithful characteristic subgroup of $$K$$. Then, $$H$$ is a coprime automorphism-faithful characteristic subgroup of $$G$$.