I'll answer a related question: in the $K(n)$-local stable category, $BG$ is dualizable for all finite groups $G$, moreover, each is self-dual. You can find this in Hovey and Strickland's 'Morava $K$-theories and localisation' Corollary 8.7.
More precisely, the result states that if $G$ is finite, then the $K(n)$ localized norm map: \begin{equation} L_{K(n)}\Sigma^\infty_+ BG\rightarrow F(L_{K(n)}\Sigma^\infty_+ BG, L_{K(n)} S) \end{equation} is a weak equivalence. This is because Tate cohomology lowers chromatic complexity (Hovey-Strickland 96, Kuhn 2004), so the cofiber of the localized norm map $L_{K(n)}t_G(L_{K(n)} S)^{G}$ is weakly contractible.
I'll answer a related question: in the $K(n)$-local stable category, $BG$ is dualizable for all finite groups $G$, moreover, each is self-dual. You can find this in Hovey and Strickland's 'Morava $K$-theories and localisation' Corollary 8.7.