Large indecomposable modules with many automorphisms

Over integral domains R admitting fully rigid systems of modules with R as endomorphism rings, we construct indecomposable modules of cardinality ? whose automorphism groups are as large as possible: they have cardinality 2?. Here ? denotes any infinite cardinal with |R| = ?. We also show that over certain valuation domains there exist indecomposable divisible torsion modules of arbitrarily large cardinalities ? whose automorphism groups have cardinality 2?.