This is a question in complex geometry, but even for algebraic varieties I don't know the answer:
Let $S$ be a smooth compact Kähler surface (for example a smooth complex projective surface) that is an elliptic bundle, i.e there exists a morphism with connected fibres $\varphi: S \rightarrow C$ onto a smooth projective curve $C$ such that all the fibres are isomorphic to a fixed elliptic curve $E$.
I am looking for references that describe the automorphisms of $S$ over $C$, i.e. those automorphisms $f: S \rightarrow S$ that send $\varphi$-fibres onto $\varphi$-fibres. More precisely I would like to know:
Which groups can appear?
Given such an automorphism $f$, can you describe the locus of fixed points?
Remarks:
If $S$ is torus, there is a very detailed description by Fujiki, but I don't think his arguments apply to the case where $C$ has higher genus.
If one assumes that S becomes a product after finite base change (this always holds if $S$ is projective, see Jason's remark below) one can try to use descent arguments, but I don't think that this is representative of the general situation. I am looking for arguments that do not use the existence of a (multi-)section.