Bridge theorems for group theory and formal languages
Let me first answer your subquestion: Does the literature on semiautomata ever look at "group-automata"?. The answer is yes. In his book (Automata, languages, and machines. Vol. B, Academic Press), S. Eilenberg gave a characterization of the regular languages recognized by finite commutative groups and $p$-groups. Similar results are known for finite nilpotent groups, soluble groups and supersoluble groups.

Finite groups also play an important role in the problem of finding a complete set of identities for regular expressions. An infinite complete set was proposed by John Conway and this conjecture was ultimately proved by D. Krob. There is a finite number of "basic" identities, plus an identity for each finite simple group. See my answer to this question for references.

In the opposite direction, finite automata theory lead to an elementary proof of basic results on combinatorial group theory, like the Schreier formula. Based on Stallings's seminal paper Topology of Finite Graphs.

Also in the opposite direction, automatic groups are defined in terms of finite automata.

Profinite groups also play an important role in automata theory. An example is the characterization of the regular languages recognized by transition-reversible automata with 
possibly several initial and final states.

For a very nice connection between context-free languages, groups and logic, see the article 
by David E. Muller and Paul E. Schupp, Context-free languages, groups, the theory of ends, second-order logic, tiling problems, cellular automata, and vector addition systems.