Complements to abelian normal subgroup are automorphic

Statement
Suppose $$G$$ is a group, $$N$$ is an Abelian normal subgroup, and $$H, K $$ are permutable complements to $$N$$ in $$G$$. Then, there is an automorphism $$\sigma$$ of $$G$$ such that:


 * The restriction of $$\sigma$$ to $$N$$ is the identity map on $$N$$.
 * $$\sigma$$ induces an isomorphism from $$H$$ to $$K$$.

In particular, $$H$$ and $$K$$ are automorphic subgroups.

(Note: There may exist automorphic subgroups to $$H$$ that are not permutable complements to $$N$$).

Related facts

 * Complement to normal subgroup is isomorphic to quotient
 * Hall retract implies order-conjugate: This is the conjugacy part of the Schur-Zassenhaus theorem, and states that, for a normal Hall subgroup, any two complements are not just automorphic, they are also conjugate subgroups.
 * Complements to normal subgroup need not be automorphic

Breakdown when the normality constraint is removed or shifted

 * Every group of given order is a permutable complement for symmetric groups: A complete breakdown of the analogous statement when the subgroup is no longer assumed to be normal.
 * Retract not implies every permutable complement is normal
 * Retract not implies normal complements are isomorphic

Other forms of breakdown

 * Semidirect product is not left-cancellative for finite groups