Corollary of Timmesfeld's replacement theorem for abelian subgroups

Statement
Suppose $$P$$ is a group of prime power order. Let $$\mathcal{A}(P)$$ denote the set of abelian subgroups of maximum order in $$P$$. If $$A \in \mathcal{A}(P)$$, and $$V$$ is an $$A$$-invariant abelian subgroup of $$P$$. Then, $$[V,A]C_A([V,A])$$ is an abelian subgroup of maximum order.

Related facts

 * Timmesfeld's replacement theorem
 * Corollary of Timmesfeld's replacement theorem for elementary abelian subgroups