It is well known that inverse semigroups model etale groupoids and their C*-algebras. We will explain how they can also be used to model their actions on C*-algebras. An inverse semigroup S can also act on a groupoid G and such an action induces one on C*(G).

The iterated crossed product theorem implies that the full crossed product C*(G)xS is canonically isomorphic to C*(GxS). Although this result looks plausible, its proof is highly non-trivial. And surprisingly, an analogous reduced version does not hold in general. This is due to exactness problems of the groupoids involved.