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We prove structural theorems for computing the completion of a G-spectrum atthe augmentation ideal of the Burnside ring of a finite group G. First we showthat a G-spectrum can be replaced by a spectrum obtained by allowing onlyisotropy groups of prime power order without changing the homotopy type of thecompletion. We then show that this completion can be computed as a homotopycolimit of completions of spectra obtained by further restricting isotropy toone prime at a time, and that these completions can be computed in terms ofcompletion at a prime.  As an application, we show that the spectrum of stable maps from BG to theclassifying space of a compact Lie group K splits non-equivariantly as a wedgesum of p-completed suspension spectra of classifying spaces of certainsubquotients of the product of G and K. In particular this describes the dualof BG.

Completion of $G$-spectra and stable maps between classifying spaces