Projective modules and the homotopy classification of $(G,n)$-complexes

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 $\mathbb{Z} G$-module under the action of $\text{Aut}(G)$. We develop techniques to compute this action explicitly and use this to give an example where the action is non-trivial.