  In [B.Gruenbaum, G.C. Shephard, Spherical tilings with transitivity
properties, in: The geometric vein, Springer, New York, 1981, pp. 65-98], they
proved "for every spherical normal tiling by congruent tiles, if it is
isohedral, then the graph is a Platonic solid, an Archimedean dual, an n-gonal
bipyramid (n>2), or an n-gonal trapezohedron (i.e., the pseudo-double wheel of
2n faces)". In the classification of spherical monohedral tilings, one
naturally asks an "inverse problem" of their result: For a spherical monohedral
tiling of the above mentioned topologies, when is the tiling isohedral? We
prove that for any spherical monohedral quadrangular tiling being topologically
a trapezohedron, if the number of faces is 6, or 8, if the tile is a kite, a
dart or a rhombi, or if the tile is convex, then the tiling is isohedral.
