  Twisted \'etale groupoid algebras have been studied recently in the algebraic
setting by several authors in connection with an abstract theory of Cartan
pairs of rings. In this paper, we show that extensions of ample groupoids
correspond in a precise manner to extensions of Boolean inverse semigroups. In
particular, discrete twists over ample groupoids correspond to certain abelian
extensions of Boolean inverse semigroups and we show that they are classified
by Lausch's second cohomology group of an inverse semigroup. The cohomology
group structure corresponds to the Baer sum operation on twists.
  We also define a novel notion of inverse semigroup crossed product,
generalizing skew inverse semigroup rings, and prove that twisted Steinberg
algebras of Hausdorff ample groupoids are instances of inverse semigroup
crossed products. The cocycle defining the crossed product is the same cocycle
that classifies the twist in Lausch cohomology.
