On the Anderson Dual of $H\underline{\mathbb Z}$ and $H\underline{\mathbb F}_2$

In this paper, we give complete classifications of the finite groups $G$ for which the constant Eilenberg--Mac Lane spectra $H\underline{\mathbb Z}$ and $H\underline{\mathbb F}_2$ are Anderson self-dual up to a virtual representation shift, and we determine all such shifts. For both of these spectra, we determine the representation periodicity lattice for every finite group. As applications, we obtain $RO(G)$-graded universal coefficient theorems relating equivariant homology and cohomology, together with formulas for the Anderson duals of $H\underline{\mathbb Z}$-modules. In the $H\underline{\mathbb F}_2$ self-dual cases, we identify the graded $\mathbb F_2$-linear dual of the coefficient ring $π_\star^G H\underline{\mathbb F}_2$ with a representation shift of the ring as a module over itself. This gives perfect multiplication pairings and graded self-injectivity.