Coprime automorphism-faithful characteristicity is transitive

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Statement

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 $G$.

Definitions used

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Proof

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