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)$?

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 $H_{q}(H(\mathbb{Z}), \mathbb{Z}/2)$?