  The study of the entanglement dynamics plays a fundamental role in
understanding the behaviour of many-body quantum systems out of equilibrium. In
the presence of a globally conserved charge, further insights are provided by
the knowledge of the resolution of entanglement in the various symmetry
sectors. Here, we carry on the program we initiated in [Phys. Rev. B 103,
L041104 (2021)], for the study of the time evolution of the symmetry resolved
entanglement in free fermion systems. We complete and extend our derivations
also by defining and quantifying a symmetry resolved mutual information. The
entanglement entropies display a time delay that depends on the charge sector
that we characterise exactly. Both entanglement entropies and mutual
information show effective equipartition in the scaling limit of large time and
subsystem size. Furthermore, we argue that the behaviour of the charged
entropies can be quantitatively understood in the framework of the
quasiparticle picture for the spreading of entanglement, and hence we expect
that a proper adaptation of our results should apply to a large class of
integrable systems. We also find that the number entropy grows logarithmically
with time before saturating to a value proportional to the logarithm of the
subsystem size.
