Frattini-embedded normal subgroup

Definition with symbols
A normal subgroup $$N$$ of a group $$G$$ is termed Frattini-embedded or Frattini-imbedded if for every proper subgroup $$H$$ of $$G$$, $$NH \ne G$$. When $$G$$ is a finite group, or more generally, when every proper subgroup of $$G$$ is contained in a maximal subgroup, then this condition is equivalent to saying that $$N$$ is contained in the Frattini subgroup of $$G$$.

Left transiter
Any characteristic subgroup of a Frattini-embedded normal subgroup is Frattini-embedded normal. It's not clear whether characteristicity is precisely the left transiter of the property of being Frattini-embedded normal.