Class-preserving implies linearly extensible

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.