Mac Lane Method for Determination of Extensions of Finite Groups. II. An Example for the Dihedral Group D$\text{}_{2}$

The Mac Lane method of classification and construction of all extensions of a group Q by an Abelian group T is demonstrated on the case Q = D$\text{}_{2}$, T = C$\text{}_{2}$. Constructions involving free groups and operator homomorphisms are performed in detail, and the complete list of resulting extensions is given. It is shown that there are 8 classes of gauge equivalency, and they fall into 4 classes of isomorphism. The role of gauge transformations is pointed out. Physical contexts of various constructions are reviewed. A comparison with the direct cohomology definitions is performed.