The absolute Galois group of the rational number field is, of course, a central object in number theory.  However, it is known to be deficient in some respects.  In 1951, André Weil defined what came to be known as the Weil group.  This is a topological group refining the Galois group: it surjects onto the absolute Galois group with nontrivial connected kernel.  The Weil group provides an extension of the theory of Galois representations, allowing for a closer connection with automorphic forms.  In this course, I will explain that there remain further deficiencies of the Weil group, which must be corrected by a further refinement.  Our motivation comes from cohomological considerations, and the refinement we discuss is homotopy-theoretic in nature and goes in an orthogonal direction from the conjectural refinement proposed by Langlands (known as the Langlands group).  Yet, as we will explain, it does have relevance for the Langlands program.