Nonstandard Cayley automatic representations of fundamental groups

We construct a new family of Cayley automatic representations of semidirect products ${\mathbb{Z}}^n{⋊}_A\mathbb{Z}$ for which none of the projections of the normal subgroup ${\mathbb{Z}}^n$ onto each of its cyclic components is finite automaton recognizable. For $n=2$ we describe a family of matrices from $\mathrm{G}\mathrm{L}(2,\mathbb{Z})$ corresponding to these representations. We are motivated by a problem of characterization of all possible Cayley automatic representations of these groups.