Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini

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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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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 must also satisfy the second group property
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Suppose G is a Frattini-embedded normal-realizable group: in other words, G can be embedded in some group H as a Frattini-embedded normal subgroup. Then G satisfies the following condition:

The inner automorphism group of G is a Frattini-embedded normal subgroup of the automorphism group of G.


Proof outline

Using notation as above, let G be the group and H be a group in which G is embedded as a Frattini-embedded normal subgroup. Then:

In case Aut(G) or H is a finite group, we can in fact use this to conclude that Inn(G) is contained in the Frattini subgroup of Aut(G).