Invariable generation and the Chebotarev invariant of a finite group

A subset S of a finite group G invariably generates G if G=〈sg(s)|s∈S〉 for each choice of g(s)∈G, s∈S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in Kowalski and Zywina (2010) [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements.