Maximal among abelian characteristic subgroups may be multiple and isomorphic

Statement
It is possible to have a group of prime power order $$P$$ with two distinct subgroups $$H, K \le P$$, such that both $$H$$ and $$K$$ are fact about::maximal among abelian characteristic subgroups, and $$H \cong K$$.

Facts used

 * 1) uses::Classification of abelian subgroups of maximum order in unipotent upper-triangular matrix groups

Related facts
The example described here also shows many other things:


 * Semidirect product is not left-cancellative for finite groups

Example involving the upper triangular matrices
Suppose $$p$$ is any prime, and let $$G := U(5,p)$$ be the group of upper-triangular unipotent $$5 \times 5$$ matrices over the field of $$p$$ elements. Let $$P$$ be the subgroup of $$G$$ comprising those matrices where the $$(12)^{th}$$ entry is zero. Then, $$P$$ is a group of order $$p^9$$.

By fact (1), we have that $$G$$ has two subgroups that are Abelian of maximum order: the rectangle groups of dimensions $$2 \times 3$$ and $$3 \times 2$$ respectively. Call these subgroups $$H$$ and $$K$$ respectively. Then, observe that:


 * Both $$H$$ and $$K$$ are also Abelian subgroups of maximum order in $$P$$. Moreover, they are the only Abelian subgroups of maximum order in $$P$$ since they are the only Abelian subgroups of maximum order in $$G$$.
 * $$H$$ and $$K$$ are isomorphic -- in fact, they are conjugate subgroups inside the bigger group $$GL(5,p)$$. This conjugation restricts to an automorphism of $$G$$, but not of $$P$$.
 * Both $$H$$ and $$K$$ are normal in $$G$$, and hence in $$P$$. The quotient $$P/H$$ is isomorphic to the unipotent subgroup of 3-by-3 matrices while the quotient $$P/K$$ is isomorphic to the elementary Abelian group of order $$p^3$$. Hence, $$H$$ and $$K$$ are not automorphic subgroups in $$P$$.
 * Thus, $$H$$ and $$K$$ are the only Abelian subgroups of their order, and they are not automorphic in $$P$$. Hence, they are both characteristic subgroups. Since they are Abelian of maximum order, they are both maximal among Abelian characteristic subgroups.

We have thus established all the required conditions.