When do lazy and invertible cellular automata generate all cellular automata?

Let $G$ be a group and let $A$ be a set with at least two elements. A lazy cellular automaton over $A^G$ acts as the identity except at the cells where a fixed pattern occurs, where it replaces the central symbol by a fixed different symbol. Let $M(G;A)$ be the submonoid of $\mathrm{CA}(G;A)$ genera