Nilpotent and every abelian characteristic subgroup is central implies class at most two

Statement

If $G$ is a Nilpotent group (?) and satisfies the property that every Abelian characteristic subgroup of it is central, then the nilpotence class of $G$ is at most two.

Facts used

1. Penultimate term of lower central series is Abelian in nilpotent group of class at least three

Proof

The proof follows direct from fact (1): it shows that in a nilpotent group of class at least three, we have an Abelian characteristic subgroup that is not central (namely, the penultimate term of the lower central series).