G-automata, counter languages and the Chomsky hierarchy

We consider how the languages of $G$-automata compare with other formal language classes. We prove that if the word problem of a group $G$ is accepted by a machine in the class $\mathcal{M}$ then the language of any $G$-automaton is in the class $\mathcal{M}$. It follows that the so called {\emph counter languages} (languages of ${\mathbb{Z}}^n$-automata) are context-sensitive, and further that counter languages are indexed if and only if the word problem for ${\mathbb{Z}}^n$ is indexed.