Relatedly, I'm
wondering whether it's possible to have a commutative, cocommutative Hopf algebra in a symmetric monoidal category whose $p$-power map is trivial for some $p$ and is dualizable but not self-dual. $\Sigma^\infty_+ G$ is dualizable for a compact Lie group $G$, but the only commutative ones are $(S^1)^{\times n}$ and finite $G$.