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We define and study the notion of a crossed module over an inverse semigroupand the corresponding $4$-term exact sequences, called crossed moduleextensions. For a crossed module $A$ over an $F$-inverse monoid $T$, we showthat 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$H^3_\le(T^1,A^1)$.

Inverse semigroup cohomology and crossed module extensions of semilattices of groups by inverse semigroups