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Due to work of Stanley Kochman in "Integral cohomology operations. Current trends in algebraic topology, Part 1 (London, Ont., 1981), pp. 437–478, CMS Conf. Proc., 2, Amer. Math. Soc., Providence, R.I., 1982. ", there is a description of the mod two homology of the integral Eilenberg Maclane spectrum. It is the $\mathbb{Z}/2$- polynomial algebra $P(\xi_1 ^2, \bar{\xi_2}, \bar{\xi_3}\ldots)$, where the bar denotes the conjugation in the sense of the canonical anti-involution in the dual of the steenrod algebra.

Is there somewhere a computation of the group cohomology of the group of order two $\mathbb{Z}/2$ with nontrivial coefficients given by the conjugation action in the mod two homology groups $H_{q}(H(\mathbb{Z}), \mathbb{Z}/2)$?

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    $\begingroup$ for the dual Steenrod algebra itself it is an unsolved problem (as far as I know) to even compute H^0, let alone the rest. $\endgroup$
    – Dylan Wilson
    Commented Aug 23, 2019 at 21:23
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    $\begingroup$ Since for any $Z/2[Z/2]$-module $M$ there is a splitting $M=F\oplus T$, computing $H^0$ is actually equivalent to computing the rest. $\endgroup$
    – user43326
    Commented Aug 24, 2019 at 6:53

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