  In this paper we characterize permutation groups that are automorphism groups
of coloured graphs and digraphs and are abelian as abstract groups. This is
done in terms of basic permutation group properties. Using Schur's classical
terminology, what we provide is characterizations of the classes of 2-closed
and $2*$-closed abelian permutation groups. This is the first characterization
concerning these classes since they were defined.
