Self-normalizing implies canonical Abelianization

Property-theoretic statement
Any self-normalizing subgroup of a group possesses a canonical Abelianization.

Verbal statement
Suppose $$B$$ is a self-normalizing subgroup of a group $$G$$. Then, the Abelianization of $$B$$ is determined upto canonical isomorphism in $$G$$. In other words, we can do the following, in a manner that is invariant under inner automorphisms:


 * Consider the space whose points are the Abelianizations of conjugates of $$B$$
 * For any two points in the space, define a unique isomorphism between those points, such that the isomorphisms form a category i.e. the isomorphism from $$p$$ to $$p$$ is the identity, and given $$p,q,r$$, the isomorphism from $$p$$ to $$r$$ is the composite of the isomorphisms from $$p$$ to $$q$$ and $$q$$ to $$r$$