Classification of Disconnected Reductive Algebraic Groups

We say a disconnected algebraic group is reductive if its connected component is a reductive group in the usual sense. Even if one is only interested in connected reductive groups, disconnected ones enter the picture as subgroups, so gaining an understanding of them is a fruitful endeavour. In this talk, I will discuss the following question: Given a connected reductive algebraic group N and a finite group H, which algebraic groups G fit into the short exact sequence 1 --> N --> G --> H --> 1? If time permits, I will also briefly discuss some representation-theoretic results for such groups, including a bound on their Knutson Index. This talk is based on the paper ``Disconnected Reductive Groups: Classification and Representations" (arXiv 2409.06375), which is joint work with Diego Martin Duro and Dmitriy Rumynin.