Analogue of Thompson transitivity theorem fails for abelian subgroups of rank two

Statement
It is possible to choose a finite group $$G$$ and a prime number $$p$$ such that $$G$$ is a group in which every p-local subgroup is p-constrained, such that:

There exists a subgroup $$A$$ of $$G$$ that is maximal among abelian normal subgroups in some $$p$$-Sylow subgroup of $$G$$, such that $$A$$ has rank two, and there is a prime $$q \ne p$$ such that $$C_G(A)$$ is not transitive on the collection of maximal $$A$$-invariant $$q$$-subgroups of $$G$$.

In other words, the analogue of the Thompson transitivity theorem fails if we drop the assumption of rank at least three.

Related facts

 * Thompson transitivity theorem
 * Analogue of Thompson transitivity theorem fails for groups in which not every p-local subgroup is p-constrained