Automaticity of the sequence of the last nonzero digits of $n!$ in a fixed base

In 2011 Deshouillers and Ruzsa tried to argument that the sequence of the last nonzero digit of $n!$ in base 12 is not automatic. This statement was proved few years later by Deshoulliers. In this paper we provide alternate proof that lets us generalize the problem and give an exact characterization in which bases the sequence of the last nonzero digits of $n!$ is automatic.