Maximal among abelian normal subgroups in p-Sylow subgroup that is not cyclic implies every invariant p'-subgroup is in the p'-core in p-constrained group

Statement
Suppose $$G$$ is a finite group that is a fact about::p-constrained group, $$p$$ is a prime number dividing the order of $$G$$. Suppose $$P$$ is a $$p$$-Sylow subgroup and $$A$$ is maximal among abelian normal subgroups in $$P$$ (hence, $$A$$ is a self-centralizing normal subgroup). Suppose further that $$A$$ has rank at least two.

Then, any $$A$$-invariant $$p'$$-subgroup of $$G$$ is contained in the p'-core $$O_{p'}(G)$$. Here, $$O_{p'}(G)$$ is the unique largest normal subgroup of $$G$$ of order relatively prime to $$p$$.