I am interested in the nonabelian finite $p$-group $G$ with the following property: 

$G$ has a maximal abelian subgroup $A$ and $x\in G\setminus A$ such that
$x$ normalize $A$ but $x$ does not commute with any noncentral element of $A$.

$p$-groups with maximal class have this property since they have maximal abelian subgroup of order $p^2$. Any reference or comment or maybe partial characterization will be useful for me.