Cocycle skew reversal generalization of Baer correspondence

Statement
This is a generalization of the Baer correspondence (see also generalized Baer correspondence) between some Lie rings of class at most two and some groups of class at most two. Specifically, it is a correspondence:

Lie ring arising as the skew of a class two near-Lie cring $$\leftrightarrow$$ group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle

In order to move back and forth between these structures, it is necessary to introduce an additional structure. This additional structure is that of a class two near-Lie cring. The additional structure choice is not unique; however, it turns out that different possible choices of the additional structure give rise to different ways of going back and forth but the group corresponding to a Lie ring remains the same up to isomorphism and vice versa.

From Lie ring to group
Suppose $$L$$ is a Lie ring arising as the skew of a class two near-Lie cring, i.e., there exists a binary operation $$*: L \times L \to L$$ such that:


 * $$x * (y + z) + (y * z) = (x + y) * z + (y * z)$$ for all $$x,y,z \in L$$, i.e., $$*$$ is a 2-cocycle for trivial group action of $$L$$ on itself.
 * $$x * y = 0$$ if $$\langle x,y \rangle$$ is cyclic.
 * $$x * (y * z) = 0$$ for all $$x,y,z \in L$$.
 * $$[x,y] = (x * y) - (y * x)$$ for all $$x,y \in L$$.

Then, we can define a group structure on $$L$$ in terms of $$*$$ as follows:

It turns out that the group commutator $$xyx^{-1}y^{-1}$$ is the same as the Lie bracket $$[x,y]$$ with these operations.

From group to Lie ring
Suppose $$G$$ is a group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle, i.e., there exists a function $$\circ:G \times G \to G$$ such that:


 * $$x \circ y \in Z(G)$$ 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 $$x,y,z \in G$$.
 * $$xyx^{-1}y^{-1} = (x \circ y)(y \circ x)^{-1}$$ for all $$x,y \in G$$.

Then we can define a Lie ring structure on $$G$$ as follows:

Particular cases
We include here some examples of finite groups of prime power order that do not fall under the Baer correspondence or the LCS-Baer correspondence but fall under this more general correspondence. This means that we only consider finite non-abelian 2-groups. Note that since any finite nilpotent group is a direct product of Sylow subgroups and the correspondence works separately on each Sylow factor, there is no loss of generality in restricting to 2-groups.

Role in explaining 1-isomorphisms
This correspondence plays an important role in explaining 1-isomorphisms between non-abelian groups of nilpotency class two and abelian groups. We list here some cases: