  Evaluating the quantum capacity of quantum channels is an important but
difficult problem, even for channels of low input and output dimension. Smith
and Smolin showed that the quantum capacity of the Clifford-twirl of a qubit
amplitude damping channel (a qubit depolarizing channel) has a quantum capacity
that is at most the coherent information of the qubit amplitude damping channel
evaluated on the maximally mixed input state. We restrict our attention to
obtaining upper bounds on the quantum capacity using a generalization of Smith
and Smolin's degradable extension technique. Given a degradable channel
$\mathcal N$ and a finite projective group of unitaries $\mathcal V$, we show
that the $\mathcal V$-twirl of $\mathcal N$ has a quantum capacity at most the
coherent information of $\mathcal N$ maximized over a $\mathcal V$-contracted
space of input states. As a consequence, degradable channels that are covariant
with respect to diagonal Pauli matrices have quantum capacities that are their
coherent information maximized over just the diagonal input states. As an
application of our main result, we supply new upper bounds on the quantum
capacity of some unital and non-unital channels -- $d$-dimensional depolarizing
channels, two-qubit locally symmetric Pauli channels, and shifted qubit
depolarizing channels.
