TITLE: What are induced representations good for?
QUESTION [5 upvotes]: Can somebody give me concrete problems in finite group representation theory that can be solved using induced representations?
Yes, I understand how induced representations are defined, and I know how to compute their characters, and I know the reciprocity formula. But it bothers me that I have never once needed to used this concept to solve a concrete problem!
For example, maybe somebody can name a finite group whose character table is difficult to construct - until you start looking at representations induced from representations of its subgroups? (Note that these induced representations don't necessarily need to be irreducible to be useful - they can be direct sums of easy-to-find irreps and a hard-to-find irrep. See Jim's answer here.)
If you know any such examples, please let me know, and please don't reveal the full solutions because I would like to try to solve them myself!

REPLY [5 votes]: Nearly the only strategy available for computing the character table of most interesting groups is to work with induced representations from understandable subgroups. For example, try computing the character table of $SL_2(\mathbb{F}_q)$. There are very few permutation representations available (and note that permutation representations are also induced representations) and eventually you'll either give up or start inducing.