Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle

Definition
A group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle is a group $$G$$ satisfying the following condition: there is a function $$\circ:G \times G \to G$$ satisfying the following conditions:


 * $$x \circ y \in Z(G)$$ for all $$x,y \in G$$.
 * $$(x \circ (yz))(y \circ z) = ((xy) \circ z)(x \circ y)$$ for all $$x,y,z \in G$$.
 * $$x \circ y$$ is the identity element whenever $$\langle x,y \rangle$$ is cyclic.
 * $$x \circ (y \circ z)$$ is the identity element for all $$x,y,z \in G$$.
 * $$x \circ y = (y \circ x)^{-1}$$ for all $$x,y \in G$$.
 * $$xyx^{-1}y^{-1} = (x \circ y)^2$$ for all $$x,y \in G$$.

This is precisely the kind of group that can participate in the cocycle halving generalization of Baer correspondence.