Relation between group theory and information theory
Sadly, group structure is nearly so limited that there isn't much one can do with it to be of use in information theory, thus the literature is prone to be fairly sparse. Even Abelian groups aren't enough structure.

Even basic abstract algebra texts which have some basic coding theory applications generally provide examples using field theory or linear algebra. If there were useful examples for groups, they'd most likely be introduced there.


Hungerford, Thomas; Abstract Algebra: An Introduction 
Rotman,J.J.; A First Course in Abstract Algebra, Third Edition


As most of Rotman's work was in group theory, if anyone would have had examples there, certainly he would. Steven Roman at UCI also did a good bit of textbook writing (separately) on coding theory and field theory and has a text Fundamentals of Group Theory: An Advanced Approach (Birkhauser), but not having read it, I suspect you'll find it barren of the group theoretical work you're looking for.

In the late 90's I seem to recall some work on binary codes defined by using codes over the alphabet $ \mathbb {Z}_4$. For some background (and possible references) see J.H. van Lint's Introduction to Coding Theory, Third Edition (Springer, 1999).

Once you take the algebraic step up to even finite fields, you're far more likely to find overlap with more coding theory. For this, try taking a look at Error-Correcting Linear Codes: Classification by Isometry and Applications (Springer) by Betten, Braun, et al. which may have some scant material on applications to groups as I recall.