Suppose that we have a finite group $G$. Its classifying space $BG$ is not naturally pointed, so if we would like to consider it as a spectrum there are two possible approaches. The first is to take any point in the classifying space and let it be the basepoint when constructing the suspension spectrum; the second is to add a disjoint basepoint.
My question is: is it possible to reconstruct $G$ from these spectra? If so, how? What information do we know about $G$ if we are given one of these spectra?