In some of the articles quoted in the present pater it is proved that every group can be represented as the automorphism group of a groupoid. The groupoid constructed in [1] is a semilattice, in [2] the groupoid is a commutative monoid, in [9] it is a semigroup, and in [8] it is a commutative groupoid. In particular, in [5] M. Gould proves that every finite or countable group G is isomorphic to the automorphism group of a left-cancellative groupoid (G, *). In the paper, using the faithful actions of a group on a set X, we define binary operations * in such a way that the corresponding grupoid (X, *) is, e.g., divisible from the left, left quasigroup, commutative, cyclic or has left cancellation, idempotents, left or righ identities. Of special interest is the case in which the action of G over X is regular, for, in this case, we show that, if X is finite or countable, then G is isomorphic to the automorphism group of a left cancellation grupoid (X, *). In particular we obtain another proof of Gould's theorem using the left regular representation of a group G on itself.
