I read a number of posts here on MO, but haven't quite found an answer to the question of where the differentials in a spectral sequence come from.

I came across a differential $d^{0,1}$ on the $E_2$-page from a 2-dim to a 3-dim vector space (in an application of the cohomological Lyndon-Hochschild-Serre s.s.). I know that $E^{0,1}_{\infty}=\mathrm{ker} d^{0,1}$.

A priori, the kernel has dimension 0, 1 or 2. How can I determine the dimension?

In general, my question is, whether there is a way of tracking down differentials in a s.s. explicitly. Where do they come from in the first place? In the books I have read, differentials seem to be given... Still, I imagine, somewhere hidden in the proof must be a construction that proves they exist and are determined. After all, in my calculation, even if 0, the kernel of my differential does not get lost, but does get promoted to star in the second cohomology group, rather than the first. And this surely has to be prevented by a properly converging s.s.

(And just out of curiosity: How would such an explicit construction be possible in the Grothendieck s.s.?)

Thank you for any insights.