  We expand on Nekov\'a\v{r}'s construction of the plectic half transfer to
define a plectic Galois action on Hilbert modular varieties. More precisely, we
study in a unifying fashion Shimura varieties associated to groups that differ
only in the centre from $R_{F/\mathbb{Q}}{\rm GL}_2$. We define plectic Galois
actions on the CM points and on the set of connected components of these
Shimura varieties, and show that these two actions are compatible. This extends
the plectic conjecture of Nekov\'a\v{r}--Scholl.
