Centerless and maximal in automorphism group implies every automorphism is normal-extensible

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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., centerless group that is maximal in its automorphism group) must also satisfy the second group property (i.e., group in which every automorphism is normal-extensible)
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Statement

Verbal statement

If a Centerless group (?) is a Maximal subgroup (?) in its Automorphism group (?), then every automorphism of the group is normal-extensible.