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 group {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 by {H^{2}(G,C(A))}.

We define and study the notion of a crossed module over an inverse semigroup and the corresponding 4-term exact sequences, called crossed module extensions. For a crossed module A over an F -inverse monoid T , we show that equivalence classes of admissible crossed module extensions of A by T are in a one-to-one correspondence with the elements of the cohomology group H 3 ≤ (T 1 , A 1 ).Contents 10 4. From H 3 ≤ (T 1 , A 1 ) to E(T, A) 13 4.1. Normalized order-preserving 3-cocycles 13 4.2. The E-unitary cover through a free group 15 4.3. From c ∈ Z 3 ≤ (T 1 , A 1 ) to a crossed module extension of A by T 16 4.4. From cohomologous c, c ′ ∈ Z 3 ≤ (T 1 , A 1 ) to equivalent crossed module extensions 27 5. The correspondence between H 3 ≤ (T 1 , A 1 ) and E ≤ (T, A) 30 5.1. From c ∈ Z 3 ≤ (T 1 , A 1 ) to a crossed module extension and back again 30 5.2. From a crossed module extension to c ∈ Z 3 ≤ (T 1 , A 1 ) and back again 32 Acknowledgments 36 References 36

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