  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.
