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

Statement

Suppose $H_{1},H_{2}\leq G$ are Intermediately isomorph-conjugate subgroup (?)s, such that $\langle H_{1},H_{2}\rangle$ is a Nilpotent group (?). Then, $\langle H_{1},H_{2}\rangle$ is also intermediately isomorph-conjugate.

Related facts

Similar facts

• Nilpotent join of pronormal subgroups is pronormal

Facts used

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

Proof

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