A
(
G
,
n
)
–complex is an
n
–dimensional CW–complex with fundamental group
G
and whose universal cover is
(
n
−
1
)
–connected. If
G
has periodic cohomology then, for appropriate
n
, we show that there is a one-to-one correspondence between the homotopy types of finite
(
G
,
n
)
–complexes and the orbits of the stable class of a certain projective
Z
G
–module under the action of
Aut
(
G
)
. We develop techniques to compute this action explicitly and use this to give an example where the action is nontrivial.