  For a group $G$, a weak Cayley isomorphism is a bijection $f:G \to G$ such
that $f(g_1g_2)$ is conjugate to $ f(g_1)f(g_2)$ for all $g_1,g_2 \in G$. They
form a group $\mathcal W(G)$ that is the group of symmetries of the weak Cayley
table of $G$. We determine $\mathcal W(G)$ for each of the seventeen wallpaper
groups $G$, and for some other crystallographic groups.
