Induction theorems for finite-dimensional complex representations of infinite groups
I believe that all examples must come from finite quotients:
A collection of subgroups is necessarily good if the element $1\in K_0(G)$ is in the subgroup generated by the images of the induction maps $K_0(H)\to K_0(G)$. That's because these images are ideals, using the formula $V\otimes Ind(W)=Ind(Res(V)\otimes W)$.
Therefore every good collection contains a finite good collection. And of course for any collection of finite index subgroups $H_i$ there is a finite index normal subgroup $N$ of $G$ that is contained in them all.
And then the collection ${H_i}$ will be good in $G$ if and only if the collection $H_i/N$ is good in the finite group $G/N$. (This last requires a little argument splitting representations of $G$, or $H_i$, into the part fixed by $N$ and its orthogonal complement and noting that this splitting is compatible with induction.)