I have a question concerning classical invariant theory. Consider binary $n$-forms (i.e. all homogeneous polynomials of degree $n$ of two variables) over the field of complex numbers. Clearly, the group $GL(2,C)$ acts on the space of all such forms by changes of the variables. A classical relative invariant is a polynomial function $I$ in the coefficients of the form such that under the $GL(2,C)$ action the value of $I$ changes only by multiplication by $\det C$ to some power $k$ ($k$ is called the weight of $I$). One can now form rational absolute invariants by taking ratios of relative invariants of equal weights.
My question is: $GL(2,C)$-orbits of what forms can be distinguished by such rational absolute invariants? How about forms with non-zero discriminant, for example? I have found some classical results by Clebsch of the 19th century and a result by Olver of 1990, but they do not quite give the result that I want. Also, Geometric Invariant Theory seems to deal only with $SL(2,C)$-actions. For $SL(2,C)$-actions the orbits can be distinguished just by polynomial invariants, but this is a completely different situation.
In some cases (e.g. for quintics) I can prove what I need, but I am wondering if there is perhaps a general result.