We study the classification problem of crossed-product
C
∗
{C^ * }
-algebras of the form
C
r
∗
(
G
)
×
α
χ
Z
C_r^ * (G){ \times _{{\alpha _\chi }}}{\mathbf {Z}}
, where
G
G
is a discrete group,
χ
\chi
is a one-dimensional character of
G
G
, and
α
χ
{\alpha _\chi }
is the unique
∗
*
-automorphism of
C
r
∗
(
G
)
C_r^ * (G)
such that if
U
U
is the left regular representation of
G
G
, then
α
χ
(
U
g
)
=
χ
(
g
)
U
g
{\alpha _{\chi }(U_{g})=\chi (g)U_{g}}
,
g
∈
G
g \in G
. When
G
=
F
n
{G = F_{n}}
, the free group on
n
n
generators, we have a complete classification of these crossed products up to
∗
*
-isomorphism which generalizes the classification of irrational and rational rotation
C
∗
{C^ * }
-algebras. We show that these crossed products are determined by two
K
K
-theoretic invariants, that these two invariants correspond to two orbit invariants of the characters under the natural
Aut
(
F
n
)
\operatorname {Aut} ({F_n})
-action, and that these two orbit invariants completely classify the characters up to automorphisms of
F
n
{F_n}
. The classification of crossed products follows from these results. We also consider the same problem for
G
G
some other groups.