Given an omega spectrum $E$, there is a type of chern character map given by its rationalization $$r:E\to E\wedge M\mathbb{R}\;,$$ where $M\mathbb{R}$ denotes a Moore spectrum. The cofiber of the map $$\mathbb{S}\to M\mathbb{R}$$ can be identified with an $M\mathbb{R}/\mathbb{Z}$ and smashing sequence on the left by $E$ yields a Bockstein sequence $$E\overset{r}{\to} E\wedge M\mathbb{R} \to E \wedge M\mathbb{R}/\mathbb{Z}\overset{\beta}{\to}\Sigma E\;.$$

Its not hard to show that the rationalization map $r$ induces a morphism of corresponding Atiyah-Hirzebruch spectral sequences. I was wondering if the Bockstein map $\beta$ did the same thing. When I sat down to try to prove it, I realized that it can't induce a full morphism of spectral sequences since it is not well defined on the $E_1$ page, but I was hoping to show that it commutes with the differential on the $E_2$ page and that the induced maps on higher pages also commute with the differentials. Does anyone know if this is true, or know of a reference where this is discussed?

Edit

I realized that I should be a bit more clear about what I'm looking for. The map which I called $\beta$ above certainly induces a map of spectral sequences. In practice though its usually trivial. I'm trying to show that the Bockstien map on the corresponding coefficients induces a morphism of spectral sequences.