If you consider the symmetry group of the regular icosahedron to be "particularly large" (the group contains several rotations about non-coplanar axes and also orientation-reversing reflections), then the following sequence of refinements of the icosahedral triangulation may be of interest to you. Each of them consists of a family of triangles, their mesh converges to zero, while the triangles remain fairly robust, *i.e.*, not too different from being equilateral and not too different from each other. (There is little hope for finding a triangulation consisting of arbitrarily many triangles that are both *small* and *congruent* at the same time.)

Start by projecting the faces of the inscribed regular icosahedron to the sphere; get 20 congruent equilateral triangles. Then partition each triangle by ***connecting the midpoints of its sides by geodesic arcs on the sphere***. Each triangle of the first triangulation is replaced by four smaller triangles; the one in the middle remains equilateral, the other three do not. Keep refining the partition recursively. After the $n$-th partition you will have a triangulation consisting of $20n^4$ triangles, all of approximately the same small diameter, and all not too far from being equilateral. I think the difference between their shapes is not greater than it is after the first partition. The symmetry group for each of these triangulations remains unchanged.