Nilpotent Hall implies isomorph-conjugate

Statement
Any fact about::nilpotent Hall subgroup (i.e., a fact about::Hall subgroup that is also a fact about::nilpotent group)of a finite group is isomorph-conjugate: it is conjugate to any isomorphic subgroup.

Facts used

 * 1) uses::Nilpotent Hall subgroups of same order are conjugate
 * 2) uses::Sylow implies intermediately isomorph-conjugate
 * 3) uses::Hall implies join of Sylow subgroups
 * 4) uses::Nilpotent join of intermediately isomorph-conjugate subgroups is intermediately isomorph-conjugate

Related facts

 * Hall not implies isomorph-conjugate
 * Hall not implies isomorph-automorphic
 * Nilpotent Hall not implies order-conjugate

Proof using fact (1)
The proof follows directly from fact (1).

Proof using facts (2)-(4)
The proof follows directly by combining facts (2)-(4).