Finiteextensible implies classpreserving
From Groupprops
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 must also satisfy the second automorphism property
View all automorphism property implications  View all automorphism property nonimplications

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Statement
Any finiteextensible automorphism of a finite group is classpreserving automorphism.
Facts used
 Finiteextensible implies finitecharacteristicsemidirectly extensible
 Finitecharacteristicsemidirectly extensible implies linearly pushforwardable over prime field (where the prime does not divide the order of the group)
 Linearly pushforwardable implies classpreserving when the field is a classseparating field
 Every finite group admits a sufficiently large prime field
 Sufficiently large implies splitting, splitting implies characterseparating, characterseparating implies classseparating