Finite-quotient-pullbackable implies class-preserving

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., finite-quotient-pullbackable automorphism) must also satisfy the second automorphism property (i.e., class-preserving automorphism)
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Statement

Suppose $G$ is a finite group and $\sigma$ is a finite-quotient-pullbackable automorphism of $G$. Then, $\sigma$ is a class-preserving automorphism of $G$: it sends every element of $G$ to within its conjugacy class.

Related facts

• Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving: We can slightly generalize the proof technique to show that the result holds not just for finite groups but also for conjugacy-separable groups.
• Finite-extensible implies class-preserving
• Finite-extensible implies subgroup-conjugating, Extensible implies subgroup-conjugating
• Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving

Facts used

1. Finite-quotient-pullbackable implies quotient-pullbackable for representation over finite field
2. Quotient-pullbackable implies linearly pushforwardable for representation over prime field
3. Linearly pushforwardable implies class-preserving for class-separating field
4. Every finite group admits a sufficiently large field
5. Sufficiently large implies splitting, Splitting implies character-separating, Character-separating implies class-separating

Proof

By facts (1) and (2), any finite-quotient-pullbackable is linearly pushforwardable over any finite prime field. By fact (3), it suffices to show that there exists a finite prime field that is class-separating for the group. This is achieved by facts (4) and (5).