  We use the controlled algebra approach to study the problem that whether the
Farrell-Jones conjecture is closed under passage to over-groups of finite
indices. Our study shows that this problem is closely related to a general
problem in algebraic $K$- and $L$-theories. We use induction theory to study
this general problem. This requires an extension of the classical induction
theorem for $K$- and $L$- theories of finite groups with coefficients in rings
to with twisted coefficients in additive categories. This extension is
well-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 to
make some reductions for the general problem, and therefore for the finite
index problem of the Farrell-Jones conjecture.
