  The finite-size scaling properties of the quantum Ising chain with different
types of generalized defects are studied. These not only mean an alteration of
the coupling constant as previously examined, but an additional arbitrary
transformation in the algebra of observables at one site of the chain. One can
distinguish between two classes of generalized defects: those which do not
affect the finite-size integrability of the Ising chain, and on the other hand
those that destroy this property. In this context, finite-size integrability is
always understood as a synonym for the possibility to write the Hamiltonian of
the finite chain as a bilinear expression in fermionic operators by means of a
Jordan-Wigner transformation. Concerning the first type of defect, an exact
solution for the scaling spectrum is obtained for the most universal defect
that preserves the global Z_2 symmetry of the chain. It is shown that in the
continuum limit this yields the same result as for one properly chosen
`ordinary' defect, that is changing the coupling constant only, and thus the
finite-size scaling spectra can be described by irreps of a shifted u(1)
Kac-Moody algebra. The other type of defect is examined by means of numerical
finite-size calculations. In contrast to the first case, these suggest a
non-continuous dependence of the scaling dimensions on the defect parameters. A
conjecture for the operator content involving only one primary field of a
Virasoro algebra with central charge c=1/2 is given.
