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We introduce the concept of a partial abstract kernel associated to a group  G  and a semilattice of groups  A  and relate the partial cohomology groupH  3   ⁢   (  G  ,   C  ⁢   (  A  )    ){H^{3}(G,C(A))}    with the obstructions to the existence of admissible extensions of  A  by  G  which realize the given abstract kernel. We also show that if such extensions exist, then they are classified byH  2   ⁢   (  G  ,   C  ⁢   (  A  )    ){H^{2}(G,C(A))}   .

The third partial cohomology group and existence of extensions of semilattices of groups by groups