A spectrum $X$ whose cohomology is $H^*(X; \mathbb{F}_2) = A//A(n)$

Question

Computing the Adams spectral sequence is hard. Even determining the $E_2$-page can be difficult, but it can be simplified in some cases. In particular, when $H^*(X; \mathbb{F}_2) = A//A(n)$, for $A$ the mod 2 Steenrod algebra, we can use the change of rings isomorphism: $$\text{Ext}_A(A//A(n), \mathbb{F}_2) = \text{Ext}_{A(n)}(\mathbb{F}_2, \mathbb{F}_2)$$ I know that $H^*(H\mathbb{Z};\mathbb{F}_2) = A//A(0)$, $H^*(ko; \mathbb{F}_2) = A//A(1)$, and $H^*(tmf; \mathbb{F}_2) = A//A(2)$. What about the higher quotients $A//A(n)$ for $n \geq 2$?