Maximal among abelian characteristic not implies self-centralizing in nilpotent

Statement
We can have a nilpotent group with a subgroup that is maximal among abelian characteristic subgroups (in other words, it is an abelian characteristic subgroup not contained in any bigger Abelian characteristic subgroup) that is not a self-centralizing subgroup: it is properly contained in its centralizer.

Related facts

 * Maximal among abelian normal implies self-centralizing in nilpotent
 * Maximal among abelian normal implies self-centralizing in supersolvable
 * Nilpotent and every abelian characteristic subgroup is central implies class at most two
 * Thompson's critical subgroup theorem

Example of the quaternion group
Let $$Q$$ be the quaternion group. Then, the center of $$Q$$ is maximal among Abelian characteristic subgroups; however, it is far from self-centralizing, since its centralizer is the whole group.