We give theorems on the average number of unramified G extensions of curves over finite fields as the size of the finite field grows, i.e. we study the function field analogs of the moments of class groups and their non-abelian analog. These theorems come about from studying components of Hurwitz schemes, including ``unstable components'' in the sense of homological stability. We discuss some of the surprises these averages reveal, and give examples of how they lead to the realization of new structure in the number field version of these questions. This talk will include joint work with Nigel Boston.