For a finite group Gamma, we consider a Gamma symmetric autonomous Newtonian system, which is asymptotically linear at infinity and has 0 and infinity as isolated degenerate critical points of the corresponding energy function. By means of the equivariant degree theory for gradient G-maps with G = Gamma x S(1), we associate to the system a topological invariant deg (infinity) - deg (0,) which is computable up to an unknown factor due to the degeneracy of the system. Under certain assumptions, this invariant still contains enough information about the symmetric structure of the set of periodic solutions, including the existence, multiplicity and symmetric classification. Numerical example are provided for Gamma being the diheral groups D(6), D(8), D(10), D(12).