Journal of Symbolic Logic 68 (3):1044-1064 (2003)

Abstract
We show that if A and $\widehat{A}$ are automorphic via Φ then the structures $S_{R}(A)$ and $S_{R}(\widehat{A})$ are $\Delta_{3}^{0}-isomorphic$ via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets
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DOI 10.2178/jsl/1058448453
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Splitting Theorems in Recursion Theory.Rod Downey & Michael Stob - 1993 - Annals of Pure and Applied Logic 65 (1):1-106.
On the Orbits of Hyperhypersimple Sets.Wolfgang Maass - 1984 - Journal of Symbolic Logic 49 (1):51-62.
Friedberg Splittings of Recursively Enumerable Sets.Rod Downey & Michael Stob - 1993 - Annals of Pure and Applied Logic 59 (3):175-199.

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