$RO(G)$-graded cohomology of a point is self-injective

Generalizing known results for $G=C_2$ and constant $\underline{\mathbb{F}}_2$-coefficients, we show the $RO(G)$-graded Bredon cohomology of a point with constant $\underline{k}$-coefficients is self-injective whenever $G$ is a finite subgroup of $SO(3)$ and $k$ is a finite field. Self-injectivity holds more generally for the $RO(G)$-graded homotopy of the Eilenberg--MacLane spectrum associated to a $k$-linear commutative Green functor $M$, whenever the spectrum $HM$ is Anderson self-dual.