On Gr-Functors between Gr-Categories: Obstruction theory for Gr-Functors of the type (φ,f)

Each Gr-functor of the type $(\varphi,f)$ of a Gr-category of the type $(\Pi,\C)$ has the obstruction be an element $\overline{k}\in H^3(\Pi,\C).$ When this obstruction vanishes, there exists a bijection between congruence classes of Gr-functors of the type $(\varphi,f)$ and the cohomology group $H^2(\Pi,\C).$ Then the relation of Gr-category theory and the group extension problem can be established and used to prove that each Gr-category is Gr-equivalent to a strict one.