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)$?