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

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., conjugacy-separable group) must also satisfy the second group property (i.e., group in which every quotient-pullbackable automorphism is class-preserving)
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Statement

In a conjugacy-separable group, every Quotient-pullbackable automorphism (?) is a 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