Nilpotent join of intermediately isomorph-conjugate subgroups is intermediately isomorph-conjugate

Statement
Suppose $$H_1, H_2 \le G$$ are fact about::intermediately isomorph-conjugate subgroups, such that $$\langle H_1, H_2 \rangle$$ is a fact about::nilpotent group. Then, $$\langle H_1, H_2 \rangle$$ is also intermediately isomorph-conjugate.

Similar facts

 * Nilpotent join of pronormal subgroups is pronormal

Facts used

 * 1) uses::Intermediate isomorph-conjugacy is normalizing join-closed
 * 2) uses::Nilpotent implies every subgroup is subnormal
 * 3) uses::Intermediately isomorph-conjugate implies intermediately subnormal-to-normal