  Constructing all extreme instances of the set of completely positive
trace-preserving (CPTP) maps, i.e., quantum channels, is a challenging valuable
open problem in quantum information theory. Here we introduce a systematic
approach that enables us to construct exactly those extreme channels that are
covariant with respect to a finite discrete group or a compact connected Lie
group. Innovative labeling of quantum channels by group representations enables
us to identify the subset of group-covariant channels whose elements are
group-covariant generalized-extreme channels. Furthermore, we exploit
essentials of group representation theory to introduce equivalence classes for
the labels and also partition the set of group-covariant channels. As a result
we show that it is enough to construct one representative of each partition. We
construct Kraus operators for group-covariant generalized-extreme channels by
solving systems of linear and quadratic equations for all candidates satisfying
the necessary condition for being group-covariant generalized-extreme channels.
Deciding whether these constructed instances are extreme or quasi-extreme is
accomplished by solving system of linear equations. We formalize the problem of
constructing and classifying group-covariant generalized extreme channels,
thereby yielding an algorithmic approach to solving, which we express as
pseudocode. To illustrate the application and value of our method, we solve for
explicit examples of group-covariant extreme channels. With unbounded
computational resources to execute our algorithm, our method always delivers a
description of an extreme channel for any finite-dimensional Hilbert-space and
furthermore guarantees a description of a group-covariant extreme channel for
any dimension and for any finite-discrete or compact connected Lie group if
such an extreme channel exists.
