Let $\cal A$ denote the mod 2 Steenrod algebra. Can the $\mathcal{A}(2)$-module structure on $\mathcal{A}(2)//\mathcal{A}(1)$ be enriched to an $\cal A$-module structure? If so, is there a finite spectrum $X$ such that $H^\*(X) = \mathcal{A}(2)//\mathcal{A}(1)$?



Realizing $\mathcal{A}(2)//\mathcal{A}(1)$ by a finite spectrum
The $A(2)$-module structure on $A(2)//A(1)$ does not extend to an $A$-module structure. In particular, there is no spectrum $X$ with $H^\*(X; F\_2) = A(2)//A(1)$ as an $A(2)$-module.


Additively, $A(2)//A(1)$ is generated by classes $g\_i$ in degree $i$
for $i = 0, 4, 6, 7, 10, 11, 13$ and $17$. The Adem relation $Sq^4 Sq^6 = Sq^{10} + Sq^8 Sq^2$ implies $Sq^{10}(g\_0) = g\_{10}$. The Adem relation $Sq^2 Sq^8 = Sq^{10} + Sq^9 Sq^1$ implies $Sq^{10}(g\_0) = 0$. This contradicts the existence of any $A$-module structure extending the given $A(2)$-module structure.


PS: Bruner's ext code (<http://www.math.wayne.edu/~rrb/papers/>) contains a script (newconsistency) that lets you verify that a purported presentation really defines an $A(2)$-module, and tells you what is needed to extend the presentation to an $A$-module structure. It could be handy if you want to realize other $A(2)$-modules by spectra.


