$\pi_0$ of $M(2) \wedge M(2)$

Question

My motivation is trying to understand Tom Goodwillie's argument here: https://mathoverflow.net/questions/87919/difficulties-with-the-mod-2-moore-spectrum and the only thing I don't get is why $\pi_0(M(2) \wedge M(2)) = \mathbb{Z}/4$. The way I tried doing this is to look at the cofiber sequence $$M(2) \rightarrow M(2) \rightarrow M(2) \wedge M(2)$$ obtained from smashing $$S \rightarrow S \rightarrow M(2)$$ with $M(2)$. Looking at the long exact sequence in homotopy groups doesn't seem to give me $\mathbb{Z}/4$. So the question is: why is $\pi_0(M(2) \wedge M(2)) = \mathbb{Z}/4$?