The Weil group is an extension of the absolute Galois group of a number field by the connected component of the identity of its idele class group. Of course, the quotient given by the Galois group acts on stuff. Whether the bigger group naturally acts on some space is a big open problem in number theory that some people think holds the key to the Riemann hypothesis.

Tate, J. Number Theoretic Background, Proc. Symp. Pure Math. 33 (1979) 3-26.