  We construct a new family of Cayley automatic representations of semidirect
products $\mathbb{Z}^n \rtimes_A \mathbb{Z}$ 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
$\mathrm{GL}(2,\mathbb{Z})$ corresponding to these representations. We are
motivated by a problem of characterization of all possible Cayley automatic
representations of these groups.
