  We consider the amenability of groupoids $G$ equipped with a group valued
cocycle $c:G\to Q$ with amenable kernel $c^{-1}(e)$. We prove a general result
which implies, in particular, that $G$ is amenable whenever $Q$ is amenable and
if there is countable set $D\subset G$ such that $c(G^{u})D=Q$ for all $u\in
G^{(0)}$. We show that our result is applicable to groupoids arising from
partial semigroup actions. We explore these actions in detail and show that
these groupoids include those arising from directed graphs, higher rank graphs
and even topological higher rank graphs. We believe our methods yield a nice
alternative groupoid approach to these important constructions.
