Maximal among abelian characteristic not implies abelian of maximum order

Statement
It is possible to have a group of prime power order $$P$$, with a subgroup $$K$$ of $$P$$ that is maximal among Abelian characteristic subgroups in $$P$$, such that $$K$$ is not an Abelian subgroup of maximum order.

Related facts

 * Abelian not implies contained in abelian subgroup of maximum order
 * Maximal among abelian characteristic subgroups may be multiple and isomorphic
 * abelian-to-normal replacement theorem for prime exponent

Example of the quaternion group
In the quaternion group, the center $$\{ \pm 1 \}$$ is the unique maximum among abelian characteristic subgroups. However, it is not an abelian subgroup of maximum order: there are cyclic subgroups of order four that are abelian.