I've enjoyed Peter Cameron's exposition of permutation groups; most have a very CS feel and are almost exclusively devoted to solving some sort of combinatorial problem (often of real use).
He has recently written some lecture notes on a problem in synchronization(course page) that make non-trivial use of permutation groups to understand finite automata. One important tool in hardware design is a "reset word" that takes the automaton from any state and brings it back to a fixed initial state. The key point being that you send the same message no matter what state the automaton is currently in, and no matter what it ends up in the fixed initial state, ready for new commands.
The goal of the course is to make progress towards solving a conjecture that if an n-state automaton has a reset word at all, then it has a short one, that is, one of length at most (n−1)2.
He also has a nice encyclopedia about "design theory". Block designs are highly symmetric arrangements that allow for efficient and accurate statistical experiments (I believe originally in agriculture, but now widely used in many areas, especially medicine) as well as dense codes in coding theory. I first learned about them in the local cryptography seminar where they were used to give better understanding of some algebraic stream ciphers.
If you do not currently have problems you want to solve using group theory, but want to learn to solve some beautiful problems in and using group theory, then I found Butler's Fundamental Algorithms of Permutation Groups to be quite good. It uses spanning trees to solve a fundamental problem in permutation groups, and shows several very good examples of how permutation groups let you very naturally prune an exponential search tree down to something that will work like a charm in practice. The open source GAP has many of these algorithms implemented in fairly easy to read procedural language (like C, but with garbage collection). It of course also has many of the modern solutions to the same problems, so you can also see how things have improved.