Kourovka Notebook Problem 21.150

Let $G$ be an extension of a normal elementary abelian subgroup $A$ by an elementary abelian group $B\cong G/A$ such that $A$ contains an element $a$ with $C_B(a)=1$. Is it true that the rank of the subgroup $Z(\langle a,B\rangle)\cap\langle a,B\rangle'$ is at most the rank of $B$?