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? 



Reference request: mod 2 cohomology of periodic KO theory
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 $H^∗(BU(k,⋯,∞),Z\_2)$*](https://www.ams.org/journals/tran/1963-107-03/S0002-9947-1963-0151963-5/S0002-9947-1963-0151963-5.pdf), 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



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> *Adams, J. F.*, On Chern characters and the structure of the unitary group, Proc. Camb. Philos. Soc. 57, 189-199 (1961). [ZBL0103.16001](https://zbmath.org/?q=an:0103.16001).
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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.


