Second homotopy group of the mod 2 Moore spectrum Let $S/2$ be the mod 2 Moore spectrum (i.e. the cofiber of $2: S \to S$). Then multiplication by 2 acts nontrivially on this spectrum: the homotopy groups of $S/2$ are  all $\mathbb{Z}/4$-modules by a formal argument, but not $\mathbb{Z}/2$-modules. For instance, $\pi_2(S/2) = \mathbb{Z}/4$.
The long exact sequence in homotopy groups enables one to determine the homotopy groups of $S/2$ (in the range that the stable homotopy groups of spheres are known) up to extension, but the extension problems are generally nontrivial: $\pi_2(S/2)$ is a case in point (as the aforementioned exact sequence $0 \to \mathbb{Z}/2 \to \pi_2(S/2) \to \mathbb{Z}/2 \to 0$ shows). 
Is there a general technique for resolving these kinds of extension problems? I can get it in this case using some computation in the cobar complex to get the $t-s = 1$ line of the Adams spectral sequence for $S/2$ and see that the Bockstein acts nontrivially. I'm curious about a more efficient method of doing this. (Another example I had in mind was $bo \wedge \mathbb{CP}^2$: this is $bu$ by the "theorem of Reg Wood" and this is visible in the ASS, but directly computing $KO$-groups of complex projective spaces yields nontrivial extension problems, I think.)
 A: This is a comment, not an answer, I suppose.  Just a reference to Adams and Walker "On complex Stiefel manifolds''.  This follow up to Adams'  "Vector fields on spheres'' directly computes the $KO$-groups of complex projective spaces (see Theorem 2.2) by the methods of VFS, which computed the complex $K$-theory of complex and real projective spaces and the real $K$-theory of real projective spaces. There are no extension problems in sight in the calculation of $KO(\mathbf{C}P^n)$.
Aside from the obvious, multiplication by $h_0$ in the ASS, method of detection of multiplication by $2$, there aren't a whole lot of systematic methods for detecting multiplication by $2$, let alone
less simple extensions.  Massey products/Toda brackets can help.
A: 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. 
A: I just ran across this question again, and realized that the following points may be interesting.
John Rognes pointed out that a complete answer to the original question is given by the factorization
$$S/2 \overset{j}{\longrightarrow} S^1
\overset{\eta}{\longrightarrow} S
\overset{i}{\longrightarrow} S/2$$ of
$S/2 \overset{2}{\longrightarrow} S/2$, and that similarly,
$ \Sigma S/\eta  \overset{\eta}{\longrightarrow} S/\eta$ factors as
$$ \Sigma S/\eta \overset{j}{\longrightarrow} S^3
\overset{\nu}{\longrightarrow} S
\overset{i}{\longrightarrow} S/\eta .$$
Here $i$ is inclusion of the bottom cell and $j$ is projection onto the top cell.  This is easy to prove (see Lemmas 12.2 and 12.2 in "The Adams Spectral Sequence for Topological Modular Forms" by John and myself.) This easy answer fails for $\nu$ and $\sigma$:  Oka shows, in "Ring Spectra with Few Cells", Japan J. Math, 1979, Lemma 1.2, that, while $ \Sigma^d Ca \overset{a}{\longrightarrow} Ca$ extends over $j$ to $ S^{2d+1} \overset{b}{\longrightarrow} Ca$,
the obstruction to lifting this over $i$ is $jb = (1-(-1)^d)a$, which is nonzero for $a=\nu$ or $a=\sigma$.
We apply this in Lemma 1.40 of our tmf book to observe that $tm\!f/\nu$ is not a ring spectrum, since $2\nu \neq 0$ in $\pi_3tm\!f$.   One effect this has is that, even though $H^*(tm\!f/\nu)$  is a Hopf algebra, and the $E_2$-term of the Adams spectral sequence for $\pi_*(tm\!f/\nu)$ is a ring, the differentials in the Adams spectral sequence do not obey the Leibniz rule.
