  Consider equipping an alphabet $\mathcal{A}$ with a group action that
partitions the set of words into equivalence classes which we call patterns. We
answer standard questions for the Penney's game on patterns and show
non-transitivity for the game on patterns as the length of the pattern tends to
infinity. We also analyze bounds on the pattern-based Conway leading number and
expected wait time, and further explore the game under the cyclic and symmetric
group actions.
