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?



Reconstructing a group from deformed classifying space
Let $X\_+$ be a space $X$ together with a disjoint base-point. Since $\Sigma^\infty(BG\_+)=\Sigma^\infty(BG)\vee S$, where $S$ is the sphere spectrum, there's not much difference between the two spectra you propose. In Example 5.2 of Martino–Priddy's 'Stable homotopy classification of $BG^{\hat{}}\_p$' you can find non-isomorphic *finite* groups whose classifying spaces have the same suspension spectrum, so the answer to your first question in its current form is no. In that paper you can also find very interesting results concerning your last question.


