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We use the controlled algebra approach to study the problem that whether theFarrell-Jones conjecture is closed under passage to over-groups of finiteindices. Our study shows that this problem is closely related to a generalproblem in algebraic $K$- and $L$-theories. We use induction theory to studythis general problem. This requires an extension of the classical inductiontheorem for $K$- and $L$- theories of finite groups with coefficients in ringsto with twisted coefficients in additive categories. This extension iswell-known to experts, but a detailed proof does not exist in the literature.We carry out a detailed proof. This extended induction theorem enables us tomake some reductions for the general problem, and therefore for the finiteindex problem of the Farrell-Jones conjecture.

On passage to over-groups of finite indices of the Farrell-Jones conjecture