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.