Class-preserving implies linearly extensible

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., class-preserving automorphism) must also satisfy the second automorphism property (i.e., linearly extensible automorphism)
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Definition

Let $G$ be a group, and $k$ a class-determining field for it. Then, any class-preserving automorphism of $G$ is linearly extensible.

Proof

The proof combines two facts:

• Class-preserving implies linearly pushforwardable: This is the crux of the proof, and uses that the field is class-determining for the group.
• Linearly pushforwardable implies linearly extensible: This is tautological.