Reading up on spectral sequences I found out that many spectral sequences can be obtained in two different ways, either by a filtration on the object or by a filtration on the invariant to be considered.
One example is the Serre spectral sequence for a reasonable wellbehaved homotopy fiber sequence $F \rightarrow X \rightarrow B$. I think the "classical" way to obtain it is by using a skeletal filtration on $B$. But I have also seen it being built from the fiberwise Postnikov-tower in $B$-parametrized spectra, for example as explained here.
Similarly the homotopy fixed point spectral sequence for a naive $G$-spectrum $E$ can be constructed from either filtering $EG$ or considering the homotopy fixed points of the Whitehead-tower of $E$ (the latter construction was used in the Krause-Nikolaus' THH-lectures here)
Something similar happens to the Atiyah-Hirzebruch spectral sequence, as is for example remarked by Maria Yakerson in her talk on twisted K-theory. The construction using the Postnikov-tower can also be found in the first link above.
Since this phenomenon seems to be widely know to the experts, I was wondering whether there has been a unified treatment. I could not find a reference explaining under which circumstances two filtrations of this kind give rise to the same spectral sequence, so I am asking here. Thank you for your time.