Nilpotent group with torsion-free abelianization need not be torsion-free

Statement
It is possible to have a nilpotent group $$G$$ such that the abelianization of $$G$$ is a torsion-free abelian group but such that $$G$$ is not torsion-free.

Rationally powered abelianization example
See Nilpotent group with rationally powered abelianization need not be rationally powered.