For every group G, the set of its subsets forms a semiring under set-theoretical union and element-wise multiplication, and forms an involution semigroup under and element-wise inversion. We show that if the group G is finite, non-Dedekind, and solvable, neither the semiring nor the involution semigroup admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.