We introduce the notion of a crossed module over an inverse semigroup which generalizes the notion of a module over an inverse semigroup in the sense of Lausch, as well as the notion of a crossed module over a group in the sense of Whitehead and Maclane. With any crossed $S$-module $A$ we associate a $4$-term exact sequence of inverse semigroups $A \xrightarrow{i} N \xrightarrow{\beta} S \xrightarrow{\pi} T$, which we call a crossed module extension of $A$ by $T$. We then introduce the so-called admissible crossed module extensions and 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 $H^3_\le(T^1,A^1)$, whenever $T$ is an $F$-inverse monoid.

The talk is based on a joint work with Mikhailo Dokuchaev and Mayumi Makuta (both from the University of São Paulo).