Let$\newcommand{\res}{\mathrm{res}}$Let $X$ be a connected finite CW complex, and let $E$ be a bounded spectrum. For simplicity, let me assume that it has homotopy groups concentrated in degrees 0,1,2.
Then, one can consider the associated Atiyah-Hirzebruch spectral sequence to compute the cohomology groups $E^n(X)$ with $E_2$ page given by $$E_2^{p,q}=H^p(X;E^q(\mathrm{pt}))=H^p(X;\pi_{-q}E)\Rightarrow E^{p+q}(X).$$ Now, the $d_2$ differentials correspond to the $k$-invariants of $E$. More precisely, if $\psi_0:HE_0\rightarrow\Sigma^2HE_1$ denotes the first $k$-invariant of $E$, then $d_2:E_2^{p,0}\rightarrow E_2^{p+2,-1}$ is the stable cohomology operation corresponding to $\psi_0$. Likewise, if if $\psi^{res}_1:HE_1\rightarrow\Sigma^2HE_2$$\psi^{\res}_1:HE_1\rightarrow\Sigma^2HE_2$ denotes the (restriction of) the second $k$-invariant of $E$, then $d_2:E_2^{p,-1}\rightarrow E_2^{p+2,-2}$ is the stable cohomology operation corresponding to $\psi^{res}_1$$\psi^{\res}_1$.
Assuming that I completely understand the second $k$-invariant $\psi_1$ of $E$, i.e. not only its restriction $\psi^{res}_1$$\psi^{\res}_1$, is there is a similar description of the $d_3$ differential of this spectral sequence? More specifically, what would I need to know about $\psi_1$ in order to ensure that $d_3$ is trivial?
