Central nonsplit extension of the integer Heisenberg group by ${\mathbb{Z}}^2$

The quotient mapis a central nonsplit extension of the Integer Heisenberg group by ${\gamma }_3(F_2)/{\gamma }_4(F_2)\cong {\mathbb{Z}}^2$. It cannot split because the source has abelianization ${\mathbb{Z}}^2$, whereas a central splitting would give an abelianization containing an additional direct factor ${\mathbb{Z}}^2$.