  We prove new results on perfect state transfer of quantum walks on quotient
graphs. Since a graph $G$ has perfect state transfer if and only if its
quotient $G/\pi$, under any equitable partition $\pi$, has perfect state
transfer, we exhibit graphs with perfect state transfer between two vertices
but which lack automorphism swapping them. This answers a question of Godsil
(Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian
product of quotient graphs $\Box_{k} G_{k}/\pi_{k}$ is isomorphic to the
quotient graph $\Box_{k} G_{k}/\pi$, for some equitable partition $\pi$. This
provides an algebraic description of a construction due to Feder (Physical
Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.
