Suppose I have a complex representation $$ \pi:G \to GL(4), $$ (let's say $n=4$) I would like to compute the its composition with the adjoint representation of $GL(4)$ in various cases, for example (1) if $\pi$ is a sum of one dimensional characters, or (2) if $\pi$ is an induced representation, or if (3) $\pi$ is a direct sum of induced representations.
How does Ad$(\pi)$ decompose? I think this should be a matter of combinatorics/linear algebra, but I have not been able to find references or examples to learn how to do this.