  We study the four well-known capacities of a two-parameter family of qubit
Pauli channels. These are the channels which are covariant under the SO(2)
group and contain the depolarizing channel as a special case. We find exact
expressions for the classical capacity and entanglement assisted capacities,
and analytically determine the regions where the quantum capacity of the
channel vanishes. We then use a flag extension to find upper bound for the
quantum capacity and private capacity of these channels in the entire region of
parameter space and also obtain the lower bound for the quantum capacity by
calculating the single shot quantum capacity numerically. In conjunction with
previous results on depolarizing channels, our result is one step forward for
determining the capacities of the full Pauli channel.
