Let $M$ be a module over the ring $R$. Extensive use is made of Krull codimension to study further the Artinian-finitary automorphism group $\[ F_1\hbox{\rm Aut}_RM \,{=}\, \{g\,{\in}\hbox{ Aut}_RM \,{:}\, M(g - 1) \hbox{ is } R\hbox{-Artinian}\} \]$ of $M$ over $R$. Substantial progress is made where either $M$ is residually Noetherian or $R$ is commutative. There are some group-theoretic consequences of the two main structure theorems.