The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.
Ramzi, Separability in homotopical algebra, Preprint arXiv:2305.17236 (2023)
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Separable extensions of noetherian rings and finite étale morphisms of noetherian schemes give rise to separable extensions of singularity categories.
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The Galois theory of $G$-spectra and the Burnside ring
The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.
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The singularity category of a separable extension
Separable extensions of noetherian rings and finite étale morphisms of noetherian schemes give rise to separable extensions of singularity categories.