Often these extension problem are solved using *Toda brackets* (as Peter May already mentioned). I will first give the general statement, not only for spectra but also for module spectra. The statement may sound a bit complicated, but is quite useful.

Let $R$ be a strictly associative ring spectrum. Let $x\in \pi_n R$ be an element in the coefficients and denote by $Cx$ the cone of $\Sigma^n R \xrightarrow{x} R$. Then we have a long exact sequence
\begin{eqnarray*}\cdots\to\pi_* \Sigma^n R\to \pi_* R \to \pi_* Cx \to \pi_{*-1} \Sigma^n R\to \pi_{*-1}R\to\cdots\end{eqnarray*}
which splits into short exact sequences of the form
\begin{eqnarray*}0\to \pi_* R/x\pi_*R \xrightarrow{\alpha} \pi_* Cx \xrightarrow{\beta} \{\pi_{*-n-1}R\}_x\to 0\end{eqnarray*}
where $\{\pi_{*-n}R\}_x$ denotes all elements which are annihilated by $x$.

Let $y\in \pi_m R$ and $z\in \pi_k R$ be elements in the coefficients of $R$ with $xy=0$ and $yz=0$. Let $\widetilde{y} \in \pi_*Cx$ be an element with $\beta(\widetilde{y}) = y$. Let $w\in \pi_*R$ be an element such that the projection of $w$ is mapped to $\widetilde{y}z$ under $\beta$. Then $w\in \langle x,y,z\rangle$.

Back to your example: We take $R = \mathbb{S}$ and $x = 2\in\pi_0 \mathbb{S}$. We have now a short exact sequence
$$ 0 \to \pi_2\mathbb{S} \to \pi_2 \mathbb{S}/2 \to \pi_1\mathbb{S} \to 0 $$
where the outer groups are isomorphic to $\mathbb{Z}/2$ and are generated by $\eta^2$ and $\eta$, respectively. Let now $y = \eta$ and $z =2$. Lift $\eta$ to an element $\widetilde{\eta} \in \pi_2 \mathbb{S}/2$. Then the statement above tells us that $2\cdot \widetilde{\eta}$ is in the image of $\langle 2, \eta, 2 \rangle = \eta^2$. In particular, $2\cdot \widetilde{\eta}$ is non-zero. This solves the extension problem.

This technique can be applied to many cases, including $KO\wedge C\eta$. I did some calculations in the category of $TMF$-modules with this.

The problem remains how to compute Toda brackets. In general, this might be difficult, but often methods to compute $\pi_*R$ give also methods to compute the Toda brackets in $\pi_* R$. For example, Massey products in the $E^2$-term of the Adams spectral sequence converge to Toda brackets (if there are no "crossing differentials"). The Massey product $\langle 2,\eta, 2\rangle$ can, for example, be computed via cobar representatives.

Regarding references: The statement about Toda brackets and cofiber sequences follows more or less directly from the definition of Toda brackets in the framework of triangulated categories. See for example, Section 4.6 of my Thesis -- although I was a little bit lazy there with signs. Regarding the convergence of Massey products to Toda brackets one finds a statement in Kochman's *Bordism, Stable Homotopy, and Adams Spectral Sequences*. Note that he uses another definition of Toda bracket, which agrees with the one in triangulated categories in the case of triple brackets (and also else if one ignores indetermenancy) as proven in some paper by Kochman.