For surfaces S with χ(S)<g-2 and (g,n)≠(1,2), Out^{I_0}(P check Γ(S)) is naturally isomorphic to Σ_n × hat{GT}, with a faithful embedding of hat{GT} into Out^{I_0}(P hat Γ(S)).
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Automorphisms of profinite mapping class groups
For surfaces S with χ(S)<g-2 and (g,n)≠(1,2), Out^{I_0}(P check Γ(S)) is naturally isomorphic to Σ_n × hat{GT}, with a faithful embedding of hat{GT} into Out^{I_0}(P hat Γ(S)).