Dualizability of automatic algebras

We make a start on one of George McNulty's Dozen Easy Problems  : “Which finite automatic algebras are dualizable?” We give some necessary and some sufficient conditions for dualizability. For example, we prove that a finite automatic algebra is dualizable if its letters act as an abelian group of permutations on its states. To illustrate the potential difficulty of the general problem, we exhibit an infinite ascending chain _A1?_A2?_A3??${\mathbf{A}}_1?{\mathbf{A}}_2?{\mathbf{A}}_3??$ of finite automatic algebras that are alternately dualizable and non-dualizable.