  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.
