  Let $\Gamma$ be a finite $G$-vertex-transitive digraph. The in-local action
of $(\Gamma,G)$ is the permutation group $L_-$ induced by the vertex-stabiliser
on the set of in-neighbours of $v$. The out-local action $L_+$ is defined
analogously. Note that $L_-$ and $L_+$ may not be isomorphic. We thus consider
the problem of determining which pairs $(L_-,L_+)$ are possible. We prove some
general results, but pay special attention to the case when $L_-$ and $L_+$ are
both quasiprimitive. (Recall that a permutation group is quasiprimitive if each
of its nontrivial normal subgroups is transitive.) Along the way, we prove a
structural result about pairs of finite quasiprimitive groups of the same
degree, one being (abstractly) isomorphic to a proper quotient of the other.
