The question mark complex is a finite spectrum whose cohomology looks like a "question mark" (when drawn as a module over the Steenrod algebra): that is, there is an element in dimension zero $a_0$, an element $a_2 = \mathrm{Sq}^2 a_0$, and an element $a_6 = \mathrm{Sq}^4 a_2$. It can be constructed by starting with $\Sigma^{-2} \mathbb{CP}^2$, which maps to $S^2$ (thanks to the cofiber sequence $S^1 \to S^0 \to \Sigma^{-2} \mathbb{CP}^2 \to S^2$) and lifting the map $\nu: S^5 \to S^2$ to $\Sigma^{-2} \mathbb{CP}^2$ (which can be done since $\eta \nu = 0$). Then, one takes the cofiber of $S^5 \to \Sigma^{-2}\mathbb{CP}^2$.

Does anyone know any good references on this? I'd like, ideally, to see a few example computations done with it; as it is I don't really have much intuition for how to work with it.