Finite-quotient-pullbackable implies class-preserving

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