A group is sofic when every finite subset can be well approximated in a finite symmetric group. The outstanding open question about soficity (posed by Gromov) is whether every group is sofic. Helfgott and Juschenko recently argued that a celebrated group constructed by Higman is unlikely to be sofic because its soficity would imply the existence of some seemingly pathological functions. We construct variations on Higman's group and explore how Helfgott and Juschenko's arguments apply to them.