A theorem of D. F. Holt states that a nilpotent group is automatic if and only if it is virtually Abelian (Theorem 8.2.8 of [\textit{D. B. A. Epstein} et al., Word processing in groups, Jones and Bartlett, Boston (1992; Zbl 0764.20017)]). In the paper under review, the authors investigate the question of whether this theorem still holds if nilpotent groups are replaced with the larger class of soluble groups. They obtain the following Theorem: Suppose that \(G\) is an ascending HNN extension of a finitely generated torsion-free nilpotent group. Then \(G\) is automatic if and only if \(G\) is virtually Abelian. Moreover, if \(G\) is not virtually nilpotent, then the isoperimetric function for \(G\) is at least exponential, and if \(G\) is not polycyclic, the Abelianized isoperimetric function for \(G\) is also at least exponential. \textit{M. R. Bridson} and \textit{S. M. Gersten} had obtained these results in the special case where \(G\) is a split extension of a free Abelian group of finite rank by an infinite cyclic group [Q. J. Math., Oxf. II. Ser. 47, No. 185, 1-23 (1996; Zbl 0852.20031)]. The authors conjecture that a soluble group is automatic if and only if it is virtually Abelian.