  Localization phenomena of quantum walks makes the propagation dynamics of a
walker strikingly different from that corresponding to classical random walks.
In this paper, we study the localization phenomena of four-state discrete-time
quantum walks on two-dimensional lattices with coin operators as one-parameter
orthogonal matrices that are also permutative, a combinatorial structure of the
Grover matrix. We show that the proposed walks localize at its initial position
for canonical initial coin states when the coin belongs to classes which
contain the Grover matrix that we consider in this paper, however, the
localization phenomena depends on the coin parameter when the class of
parametric coins does not contain the Grover matrix.
