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