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In the section In equivariant homotopy theory I have added that the Weyl group is the automorphism group of the corresponding coset in the orbit category:
$End_{G Orbits} \big( G/H \big) \;\; = Aut_{G Orbits} \big( G/H \big) \;\; \simeq \;\; W_G(H) \,.$1 to 2 of 2