Finite-(Frattini-embedded normal)-realizable group

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.