Abstract: One variation on the theme of ``quantum symmetry" is a categorical group action on a unitary modular tensor category, which can be interpreted physically as a global symmetry of a 2-dimensional topological quantum phase of matter. Much of our understanding of tensor category theory and hence topological phases comes from categorification: from generalizing theorems we have about rings to theorems about categories. For example, categorifying an easy theorem in commutative ring theory, the work of Etingof, Nikshych, and Ostrik established an equivalence between categorical G-actions on modular tensor categories (MTCs), and so-called G-crossed braided extensions of MTCs. Physicists Barkeshli, Bonderson, Cheng, and Wang then recognized that this correspondence can be understood as a tensor-categorical formulation of gauge coupling, wherein G-crossed braided extensions of MTCs give an algebraic theory of symmetry-enriched topological (SET) phases of matter. While the abstract theory of Etingof, Nikshych, and Ostrik is well understood, even constructing the de-categorified part of G-crossed braided extensions of MTCs, namely their fusion rings, is challenging problem in general. We will give a two-part talk, starting with an introduction to the algebraic theory of SET phases described above. In the second part of the talk we describe a topological phase-inspired approach to constructing the fusion rings of certain G-crossed extensions called permutation extensions and share work in progress with E. Samperton in constructing their categorifications.