Intermediately fully invariant subgroup

Symbol-free definition
A subgroup of a group is termed intermediately fully invariant or intermediately fully characteristic if it is fully invariant in every intermediate subgroup of the group containing it.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed intermediately fully invariant or intermediately fully characteristic in $$G$$ if, for any intermediate subgroup $$K$$ of $$G$$, $$H$$ is fully characteristic in $$K$$: for any endomorphism $$\varphi$$ of $$K$$, $$\varphi(H) \le H$$.

Stronger properties

 * Weaker than::Homomorph-containing subgroup
 * Weaker than::Subhomomorph-containing subgroup
 * Weaker than::Variety-containing subgroup
 * Weaker than::Normal Sylow subgroup
 * Weaker than::Normal Hall subgroup
 * Weaker than::Normal subgroup having no nontrivial homomorphism to its quotient group

Weaker properties

 * Stronger than::Fully invariant subgroup
 * Stronger than::Intermediately characteristic subgroup
 * Stronger than::Characteristic subgroup

Metaproperties
An intermediately fully characteristic subgroup of an intermediately fully characteristic subgroup need not be intermediately fully characteristic.

An arbitrary join of intermediately fully characteristic subgroups is intermediately fully characteristic. This follows from the fact that the intermediately operator preserves the property of being closed under joins.