  Quantum walks (QWs) describe particles evolving coherently on a lattice. The
internal degree of freedom corresponds to a Hilbert space, called coin system.
We consider QWs on Cayley graphs of some group $G$. In the literature,
investigations concerning infinite $G$ have been focused on graphs
corresponding to $G=\mathbb{Z}^d$ with coin system of dimension 2, whereas for
one-dimensional coin (so called scalar QWs) only the case of finite $G$ has
been studied. Here we prove that the evolution of a scalar QW with $G$ infinite
Abelian is trivial, providing a thorough classification of this kind of walks.
Then we consider the infinite dihedral group $D_\infty$, that is the unique
non-Abelian group $G$ containing a subgroup $H\cong\mathbb{Z}$ with two cosets.
We characterize the class of QWs on the Cayley graphs of $D_\infty$ and, via a
coarse-graining technique, we show that it coincides with the class of
spinorial walks on $\mathbb{Z}$ which satisfies parity symmetry. This class of
QWs includes the Weyl and the Dirac QWs. Remarkably, there exist also spinorial
walks that are not coarse-graining of a scalar QW, such as the Hadamard walk.
