For profinite groups G acting trivially on a spectrum X, the paper gives conditions under which X^{hG} is weakly equivalent to the colimit over finite quotients of X^{hG/N}.
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A lift from group cohomology to spectra for trivial profinite actions
For profinite groups G acting trivially on a spectrum X, the paper gives conditions under which X^{hG} is weakly equivalent to the colimit over finite quotients of X^{hG/N}.