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For every group G, the set 
$\mathcal {P}(G)$ of its subsets forms a semiring under set-theoretical union 
$\cup $ and element-wise multiplication 
$\cdot $, and forms an involution semigroup under 
$\cdot $ and element-wise inversion 
${}^{-1}$. We show that if the group G is finite, non-Dedekind, and solvable, neither the semiring 
$(\mathcal {P}(G),\cup ,\cdot )$ nor the involution semigroup 
$(\mathcal {P}(G),\cdot ,{}^{-1})$ admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.
