Automorphism groups of finite groupoids

Given a finite set M of size n and a subgroup G of Sym(M), G is pertinent iff it is the automorphism group of some groupoid ??M; *??. We examine when subgroups of Sym(M) are and are not pertinent. For instance, A _n , the alternating group on M, is not pertinent for n > 4. We close by indicating a natural extension of our ideas, which relates to a question of M. Gould.