Hall conjugacy and embedding in finite soluble groups

For any prime set $\pi$, every finite soluble group has Hall $\pi$-subgroups, any two are conjugate, and every $\pi$-subgroup lies in one. Induction uses an elementary abelian minimal normal subgroup $N$ of characteristic $p$. If $p\in \pi$, the preimage of a quotient Hall subgroup is itself a Hall subgroup. If $p\notin \pi$, coprime splitting over an elementary abelian normal subgroup gives a conjugate class of complements in that preimage. The same complement conjugacy in the preimage of a smaller quotient subgroup gives the embedding assertion.