Finite-(Frattini-embedded normal)-realizable group

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:

• $N$ is contained in the Frattini subgroup of $G$
• $NH$ is a proper subgroup for any proper subgroup $H$ of $G$

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