  Two versions of the Ajtai-Szemer\'edi Theorem are considered in the Cartesian
square of a finite non-Abelian group $G$. In case $G$ is sufficiently
quasirandom, we obtain strong forms of both versions: if $E \subseteq G\times
G$ is fairly dense, then $E$ contains a large number of the desired patterns
for most individual choices of `common difference'. For one of the versions, we
also show that this set of good common differences is syndetic.
