We examine the connections between several automorphism groups associated with a saturated differentially closed field U of characteristic zero. These groups are: Γ, the automorphism group of U; the automorphism group of Γ; , the automorphism group of the differential combinatorial geometry of U and , the group of field automorphisms of U that respect differential closure.Our main results are:• If U is of cardinality λ+=2λ for some infinite regular cardinal λ, then the set of subgroups of Γ consisting of (...) all pointwise stabilizers of Dcl's of finite subsets of U is invariant under Aut .• If U is of arbitrary infinite cardinality, then each automorphism of the differential combinatorial geometry is induced by a field automorphism of U that respects differential closure .• If U is of cardinality λ+=2λ for some infinite regular cardinal λ, then each automorphism of Γ is induced by an element of acting on Γ by conjugation .• If U is of cardinality λ+=2λ for some infinite regular cardinal λ, then the outer automorphism group of Γ is isomorphic to the multiplicative group of the rationals. (shrink)