Studying group cohomology I landed on a paper with a lot of induced maps. However, I stumbled on something which seems to be a mistake, to me, although the author seems very confident with what he says (there and in the following). Essentially, it says:
We have a group extension $A\rightarrow B\rightarrow C$, where $A$ is torsion-free and $C=\langle c\rangle$, a cyclic group of order say $m$, has an automorphism $\alpha$ such that $c^\alpha=c^n$ with $1\leq n<m$ and $(m,n)=1$ (obviously). Now, the text says that $\alpha$ induces in $Ext(C,A)$ an automorphism $x\mapsto lx$ (denoting $Ext(C,A)$ additively) with $1\leq l<m$ and $ln\equiv 1$ mod $m$.
I know, on the other hand, that by the functoriality of $Ext(-,A)$, the multiplication by an integer should induce the multiplication by the same integer. Where is the problem?