Thanks, Angelo. Now I see where the equation $f$ comes in. I agree that it looks uninviting, but if no one else comes up with a better idea, this is the way to go. I would guess that calculations like this have been done 150 years ago in invariant theory.

That is an interesting example, as it both shows that the action may be nontrivial, and gives an interpretation of the fixed part. What would you say is the best way to 'check' the decomposition of $H^1(X,T_X)$ as a $G$-representation? In the given case, I would try and write that as the quotient of $H^0(X,\mathcal{O}(5))$ by $H^0(X,T_{\mathbb{P}^3}$, but this does no seem to depend on the given equation (except for its degree), and I wouldn't know the action on the latter.