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Is there any good reference for the Pontrjagin ring structure on $$ H_\ast(K(\mathbb{Z}/2,k);\mathbb{Z}/2)\cong H_\ast(\Omega K(\mathbb{Z}/2,k+1);\mathbb{Z}/2)? $$ I am familiar with Serre's theorem describing the mod 2 cohomology ring structure. I'm also aware that the action of the Dyer--Lashof operations on this infinite loop space is trivial.

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By naturality and the external Cartan formula, the standard polynomial generators of $H^*(K(\mathbf{Z}/2,k);\mathbf{Z}/2)$ given by iterated Steenrod operations on the fundamental class are primitive. Therefore the homology is a divided power algebra on the dual elements.

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