Intermediately join-transitively subnormal subgroup

Symbol-free definition
A subgroup of a group is termed intermediately join-transitively subnormal if it is a defining ingredient::join-transitively subnormal subgroup inside every intermediate subgroup.

Stronger properties
Any property stronger than the property of being a join-transitively subnormal subgroup, that also satisfies the intermediate subgroup condition, is stronger than the property of being intermediately join-transitively subnormal.