Homogeneity properties in large compact S-spaces

Under Jensen's Diamond Principle, we show how to construct a large compact S-space while having some control over its group of autohomeomorphisms. In particular we can make the space rigid or h-homogeneous (i.e. any two clopen subsets are homeomorphic). We also get a space which is B-homogeneous and not h-homogeneous, partially answering a question of M.V. Matveev. A space is B-homogeneous if it has a base every element of which can be mapped onto every other one by a homeomorphism of the whole space.