Characteristic maximal subgroups may be isomorphic and distinct in group of prime power order

Statement
It is possible to have a group of prime power order $$G$$ and two subgroups $$H$$ and $$K$$ of $$G$$ such that:


 * 1) $$H$$ and $$K$$ are both fact about::characteristic subgroups of $$G$$.
 * 2) $$H$$ and $$K$$ are both maximal subgroups of $$G$$.
 * 3) $$H$$ and $$K$$ are isomorphic groups: They are hence also fact about::series-equivalent subgroups of $$G$$, because the quotients $$G/H$$ and $$G/K$$ are both cyclic of the same prime order.
 * 4) $$H$$ and $$K$$ are distinct subgroups, i.e., they are not equal as subgroups of $$G$$.

Related facts
See series-equivalent not implies automorphic.

Proof
The smallest known example is with $$G$$ a group of order 64. There are many examples with this order, one of which has $$G$$ isomorphic to SmallGroup(64,13) and $$H$$ and $$K$$ both isomorphic to nontrivial semidirect product of Z4 and Z8 (group ID: (32,12)).