Finite-extensible implies class-preserving

Statement
Any finite-extensible automorphism of a finite group is a class-preserving automorphism.

Other results towards the associated conjecture/problem

 * Finite-extensible implies inner: Stronger results can be used to show that in fact, any finite-extensible automorphism of a group is an inner automorphism.
 * Extensible implies subgroup-conjugating
 * Finite-extensible implies subgroup-conjugating

Other facts about finite groups proved using the same method

 * Finite solvable-extensible implies class-preserving: Essentially, the same proof works, because if the original group is solvable, all the bigger groups constructed are also solvable.
 * Finite-quotient-pullbackable implies class-preserving
 * Hall-extensible implies class-preserving

Facts about infinite groups proved using similar constructions

 * Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving
 * Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving

Facts used

 * 1) uses::Finite-extensible implies semidirectly extensible for representation over finite field of coprime characteristic
 * 2) uses::Semidirectly extensible 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 finite prime field
 * 5) uses::Sufficiently large implies splitting, uses::splitting implies character-separating, uses::character-separating implies class-separating

Proof
Facts (1) and (2) combine to yield that any finite-extensible automorphism is linearly pushforwardable over a (finite) prime field where the prime does not divide the order of the group, and fact (3) yields that if the field chosen is a class-separating field for the group, then the automorphism is class-preserving. Thus, we need to show that for every finite group, there exists a prime field with the prime not dividing the order of the group, such that the field is a class-separating field for the group. This is achieved by facts (4) and (5).