Automorphism groups are studied, and variations on the small index property are proved for several kinds of structures. The main results are: 1) It is consistent with ZF to assume that every countable structure has the small index property. 2) (assuming the Continuum Hypothesis) If \({\mathfrak M}\) is a saturated stable structure of cardinality \(\aleph_ 1\) and if H is a normal subgroup of the group Aut(\({\mathfrak M})\) of automorphisms of \({\mathfrak M}\) whose index is at most \(\aleph_ 1\), then H contains every strong automorphism of \({\mathfrak M}\). 3) (with the Continuum Hypothesis) If \({\mathfrak M}\) is an \(\omega\)-stable saturated structure of cardinality \(\aleph_ 1\) and if H is a subgroup of Aut(\({\mathfrak M})\) of index at mot \(\aleph_ 1\), then there exists a countable subset A of \({\mathfrak M}\) such that every automorphism of \({\mathfrak M}\) leaving A pointwise fixed is in H. 4) The same thing is true for the dense linear ordering.