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 G is accepted by a machine in the class ℳ then the language of any G-automaton is in the class ℳ. It follows that the so called counter languages (languages of ℤⁿ-automata) are context-sensitive, and further that counter languages are indexed if and only if the word problem for ℤⁿ is indexed.