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I think that this page is highly misleading. There are various notions of G-spectra and the most common (the one used in Hill-Hopkins-Ravenel's paper on the Kervaire invariant one problem) is quite clearly not spectra with a G-action. For more details see this excellent MO answer by Tyler Lawson. I might take a crack at rewriting the article this week. DenisNardin (talk) 13:10, 5 June 2016 (UTC)