Frattini-embedded normal-realizable group

Definition
A group $$N$$ is termed Frattini-embedded normal-realizable if there exists a group $$G$$ and an embedding of $$N$$ in $$G$$ such that $$N$$ is a Frattini-embedded normal subgroup of $$G$$. In other words, $$N$$ is a normal subgroup and $$NH$$ is a proper subgroup of $$G$$ for any proper subgroup $$H$$ of $$G$$.

Stronger properties

 * Finite-Frattini-realizable group
 * Finite-(Frattini-embedded normal)-realizable group

Weaker properties

 * Inner-in-automorphism-Frattini group: This condition states that the inner automorphism group lies inside the Frattini subgroup of the automorphism group.
 * ACIC-group: