The compact real Lie group $G_2$ realizes the desuspension of $A(2)/\!/A(1)$. Recall that $$H^\ast(G_2;{\mathbb F}_2) = {\mathbb F}_2[x_3,x_5]/(x_3^4,x_5^2)$$ with $Sq^2x_3=x_5$, $Sq^1x_5 = x_3^2$. If I'm not mistaken you get an isomorphism to $A(2)/\!/A(1)$ via $$x_3\leftrightarrow Sq(4),\quad x_5\leftrightarrow Sq(0,2),\quad x_3^2\leftrightarrow Sq(0,0,1).$$

As pointed out by John Rognes, this answer is not correct (it misses the $Sq^4$ from the bottom class to the class in dimension $4$). Sorry for the confusion.

=========== previous answer ================================

The compact real Lie group $G_2$ realizes the desuspension of $A(2)/\!/A(1)$. Recall that $$H^\ast(G_2;{\mathbb F}_2) = {\mathbb F}_2[x_3,x_5]/(x_3^4,x_5^2)$$ with $Sq^2x_3=x_5$, $Sq^1x_5 = x_3^2$. If I'm not mistaken you get an isomorphism to $A(2)/\!/A(1)$ via $$x_3\leftrightarrow Sq(4),\quad x_5\leftrightarrow Sq(0,2),\quad x_3^2\leftrightarrow Sq(0,0,1).$$