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We answer a question on group connectivity suggested by Jaeger et al. [Groupconnectivity of graphs -- A nonhomogeneous analogue of nowhere-zero flowproperties, JCTB 1992]: we find that $\mathbb Z_2^2$-connectivity does notimply $\mathbb Z_4$-connectivity, neither vice versa. We use a computer to findthe graphs certifying this and to verify their properties using non-trivialenumerative algorithm. While the graphs are small (the largest has 15 verticesand 21 edges), a computer-free approach remains elusive.

Group Connectivity: $\mathbb Z_4$ v. $\mathbb Z_2^2$