Cocycle halving generalization of Baer correspondence

Statement
This is a generalization of the 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 double of a class two Lie cring $$\leftrightarrow$$ group of nilpotency class two whose commutator map is the double of a skew-symmetric 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 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.