Hall-extensible implies class-preserving

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

Hall-extensible automorphism
An automorphism $$\sigma$$ of a finite group $$G$$ is termed Hall-extensible if, for any group $$H$$ containing $$G$$ as a Hall subgroup, there is an automorphism $$\sigma'$$ of $$H$$ whose restriction to $$G$$ equals $$\sigma$$.

Class-preserving automorphism
An automorphism $$\sigma$$ of a finite group $$G$$ is termed class-preserving if it sends every element to a conjugate element.

Related facts

 * Hall-semidirectly extensible implies class-preserving
 * Finite-extensible implies class-preserving
 * Finite-quotient-pullbackable implies class-preserving

Facts used

 * 1) uses::Hall-extensible implies Hall-semidirectly extensible
 * 2) uses::Hall-semidirectly extensible implies linearly pushforwardable over prime field
 * 3) uses::Linearly pushforwardable implies class-preserving for class-separating field
 * 4) uses::Every finite group admits a sufficiently large prime field
 * 5) uses::Sufficiently large implies splitting, uses::Splitting implies character-separating, Character-separating implies class-separating

Proof
Facts (1) and (2) combine to yield that any Hall-extensible automorphism is linearly pushforwardable over a 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).