Whenever people with weak abstract algebra backgrounds ask me to explain group cohomolgy, I typically say something along the lines of: "Taking G-invariants is a very natural process. The failure of this process to preserve short exact sequence is sorrowing. To make things better, we fix the short exact sequence in a canonical way with group cohomology."
Recently, however, I've become a bit more dubious of this motivation. Given an exact sequence $0\to A^G\to B^G\to C^G$, fixing it is easy:
$$0\to A^G\to B^G\to C^G\to C^G/\text{Im}\left(B^G\to C^G\right)\to 0.$$
The question is thus
Why is the cohomology way of fixing exact sequences better than this one? How should I motivate/explain to a beginner that it is important that the term added on the right only depends on the $G$-module structure of $A$, and not on $B$ or $C$?