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

Statement
It is possible to choose a finite group $$G$$ and a prime number $$p$$ such that $$G$$ is not a group in which every p-local subgroup is p-constrained, and such that:

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

In other words, the analogue of the Thompson transitivity theorem fails if we drop the assumption that the group is a group in which every p-local subgroup is p-constrained.

Related facts

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