Certainly for the finite groups of Lie type there is a general technology for treating induced characters from parabolics, based on Deligne-Lusztig theory. Even without getting fully into that story, Lusztig's methods yield (recursively) the degrees of irreducible characters. The 1985 book by Roger Carter Finite Groups of Lie Type develops most of the theory up to that time, though later papers by Lusztig and others refine the results in many directions. See for example Chapter 10 of Carter's book for the decomposition of induced characters in the Hecke algebra spirit. (Some of this was worked out earlier by people including Curtis, KilmoyerIwahori, Lehrer Kilmoyer in the important special case of the character induced from the trivial character of a Borel subgroup in the finite group, or parabolic analogues.)
Small rank groups had been treated earlier in some detail, as noted in the question for
$Sp_4$, but the general methods lead much farther than the ad hoc methods used in early papers.
It should be emphasized that general methods like Mackey theory for finite groups stop well short of solving these kinds of problems for groups of Lie type. Even the original Deligne-Lusztig paper has to be supplemented by further work on Hecke algebra methods for decomposition of induced characters. But eventually this all becomes a unified theory for these particular families of groups. (A short treatment by Digne-Michel is also given in their LMS Student Text softcover book, but with less concrete detail than Carter gives.)