Assuming OCA and Martin's axiom for sigma-linked posets, all isomorphisms between metric reduced products of symmetric groups are trivial, and the outer automorphism group is described explicitly.
Ward Henson, and Alexander Usvyatsov, Model theory for metric structures , Model theory with applications to algebra and analysis
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On automorphism groups of metric reduced products of symmetric groups
Assuming OCA and Martin's axiom for sigma-linked posets, all isomorphisms between metric reduced products of symmetric groups are trivial, and the outer automorphism group is described explicitly.