Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving

Statement
In a conjugacy-separable group, every fact about::quotient-pullbackable automorphism is a fact about::class-preserving automorphism.

Related facts

 * Finite-quotient-pullbackable implies class-preserving
 * Finite-extensible implies class-preserving
 * Extensible implies subgroup-conjugating, finite-extensible implies subgroup-conjugating
 * Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving