TITLE: Reference request: mod 2 cohomology of periodic KO theory
QUESTION [7 upvotes]: The  mod  2 cohomology  of  the  connective ko spectrum  is  known  to  be   the  module $\mathcal{A}\otimes_{\mathcal{A}_2} \mathbb{F}_{2}$, where $\mathcal{A}$  denotes  the  Steenrod  algebra, and $\mathcal{A}_2$ denotes  the  subalgebra  generated  by  $Sq^1$  and $Sq^2$. 
Where  can  I find  the   original  calculation for  referencing it ?
What  is  the  structure  as  module  over  the  steenrod  algebra  of  the  mod  2  cohomology of periodic  KO theory?

REPLY [10 votes]: Ravenel in his Complex Cobordism and Stable Homotopy Groups of Spheres attributes this result to Stong, in Determination of $H^*(BO(k,⋯,∞),Z_2)$ and $H^∗(BU(k,⋯,∞),Z_2)$, but looking at that paper (which is concerned mainly with the "unstable" cohomology of the various constituent spaces of $ko$), he attributes this further to

Adams, J. F., On Chern characters and the structure of the unitary group, Proc. Camb. Philos. Soc. 57, 189-199 (1961). ZBL0103.16001.

There you can find indeed the required result as Lemma 4.
Regarding your second question, the cohomology $H^*(KO;\mathbb{F}_2)$ is zero for chromatic reasons (the spectrum $KU\wedge H\mathbb{F}_2$ carries an isomorphism of the additive and multiplicative formal group law in characteristic two, so it must be trivial, and then you can use the fact that $KU=KO/\eta$ to conclude, since $\eta$ is nilpotent in $\pi_*KO$), hence the map $H^*(KO;\mathbb{F}_2)→H^*(ko;\mathbb{F}_2)$ is trivial.