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This paper is inspired by the PQ penny flip game. It employs group-theoreticconcepts to study the original game and also its possible extensions. We showthat the PQ penny flip game can be associated with the dihedral group $D_{8}$.We prove that within $D_{8}$ there exist precisely two classes of winningstrategies for Q. We establish that there are precisely two different sequencesof states that can guaranteed Q's win with probability $1.0$. We also show thatthe game can be played in the all dihedral groups $D_{8 n}$, $n \geq 1$, withany significant change. We examine what happens when Q can draw his moves fromthe entire $U(2)$ and we conclude that again, there are exactly two classes ofwinning strategies for Q, each class containing now an infinite number ofequivalent strategies, but all of them send the coin through the same sequenceof states as before. Finally, we consider general extensions of the game withthe quantum player having $U(2)$ at his disposal. We prove that for Q to surelywin against Picard, he must make both the first and the last move.

The connection between the $PQ$ penny flip game and the dihedral groups