In this paper, we use the notion of twisted subgroups (i.e. subsets of group elements closed under the binary operation (a,b) aba) to provide the first structural characterization of optimal play in the Explorer-Director game, introduced as the Magnus-Derek game by Nedev and Muthukrishnan and generalized to finite groups by Gerbner. In particular, we reduce the game to the problem of finding the largest proper twisted subgroup, and as a corollary we resolve the Explorer-Director game completely for all nilpotent groups.