A partial abstract kernel of a semilattice of groups by a group admits an admissible extension exactly when its obstruction in H^3 vanishes, and if it does, the extensions are classified by H^2(G,C(A)).
Partial cohomology of groups and extensions of semilattices of abelian groups
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The third partial cohomology group and existence of extensions of semilattices of groups by groups
A partial abstract kernel of a semilattice of groups by a group admits an admissible extension exactly when its obstruction in H^3 vanishes, and if it does, the extensions are classified by H^2(G,C(A)).