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 Characteristic subgroup (?)s of $G$.
2. $H$ and $K$ are both maximal subgroups of $G$.
3. $H$ and $K$ are isomorphic groups: They are hence also 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#Related facts.

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)).