Some computations with Motivic Cohomology Spectral Sequence

We are trying to find some explicit expressions for differentials in the Motivic Cohomology Spectral Sequence(MCSS).
The behavior of differentials in the MCSS is quite similar to the topological case. Being taken with rational coefficients the sequence collapses at its E2-term.
On the other hand, its structure with integer coefficients becomes too tangled, because of the interrelation of different p-prime effects involved.
In order to avoid this entanglement, we consider MCSS with p-local coefficients (all primes but p are inverted). In this case one gets non-trivial differentials
of rather high degree and that makes their computation an interesting question.