Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic

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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., Frattini-embedded normal-realizable group) must also satisfy the second group property (i.e., group in which every automorph-conjugate subgroup is characteristic)
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This fact is related to the problem of realization related to the following subgroup-defining function: Frattini subgroup
Realization problems are usually about which groups can be realized as subgroups/quotients related to a subgroup-defining function.
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Statement

Verbal statement

Any Frattini-embedded normal-realizable group (i.e. any group that occurs as a Frattini-embedded normal subgroup of some group) must be an ACIC-group.

Related facts

A special case of this is that for a finite group, the Frattini subgroup is ACIC.

Intermediate properties

Proof