Finite-(Frattini-embedded normal)-realizable group

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This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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Definition

A finite group N is termed finite-(Frattini-embedded normal)-realizable if there exists a finite group G and an embedding of N in G such that the following equivalent conditions hold:

(note that the two conditions are not equivalent for infinite groups). The latter condition is termed being a Frattini-embedded normal subgroup.