I was learning about the applications of Group Theory and one important application is looking at the selection rules in a weak EM field. We essentially want to see whether the matrix element $\langle i|H'|f\rangle$ vanishes due to the reasons of symmetry. Here $i$ denotes the initial state, $f$ denotes the final state and $H'$ denotes the perturbed part of the hamiltonian. Dresselhaus (link: http://web.mit.edu/course/6/6.734j/www/group-full02.pdf, page 139, equation 7.29) states that the perturbed Hamiltonian can be expanded in terms of the irreducible representations of the group of the Hamiltonian.
How can we have a right to do this as the perturbed hamiltonian is not necessarily in the group of the unperturbed Hamiltonian? I understand that by Maschke's theorem we can expand any representation of group $G$ by its irreducible representations. But can an arbitrary reducible representation of a group $H$, be expanded by the irreducible representations of $G$? Furthermore, even though the irreps (irreducible representations) of $G$ form on orthogonal basis, how can we make sure that the perturbed Hamiltonian spans a space that has the same dimensionality as the irreps of the Group of the Hamiltonian?