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See Peter May's "A General Algebraic approach to Steenrod Operations". Both arise from considering the inclusion of the Sylow $p$-subgroup into $\Sigma_{p^2}$. There is a universal $\mathbb{Z}$-indexed set of operations with the Steenrod algebra and Dyer-Lashof algebra arising as the quotients acting on the homology of appropriate non-negative (resp., non-positive) complexes.