Spherical folding tessellations by kites and isosceles triangles III

The classification of the dihedral folding tessellations of the sphere and the plane whose prototiles are a kite and an equilateral triangle were obtained in 2012 by the authors. In this paper we extend this classification presenting all the dihedral folding tessellations of the sphere by kites and isosceles triangles in a particular case of adjacency that consists of two discrete families denoted by $\mathcal A^k$ and $\mathcal B^k$, $k\ge3$. A list containing these tilings including its combinatorial structure is presented.