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

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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.

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