  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
$H^3_\le(T^1,A^1)$.
