Group Connectivity: ${\mathbb{Z}}_4$ v. ${\mathbb{Z}}_2^2$

We answer a question on group connectivity suggested by Jaeger et al. [Group connectivity of graphs – A nonhomogeneous analogue of nowhere-zero flow properties, JCTB 1992]: we find that ${\mathbb{Z}}_2^2$-connectivity does not imply ${\mathbb{Z}}_4$-connectivity, neither vice versa. We use a computer to find the graphs certifying this and to verify their properties using non-trivial enumerative algorithm. While the graphs are small (the largest has 15 vertices and 21 edges), a computer-free approach remains elusive.