Let G ⊂ GL(V ) be a complex reductive group where dimV < ∞, and let π : V → V//G be the categorical quotient. Let N := ππ(0) be the null cone of V , and let H be the subgroup of GL(V ) which preserves the ideal I of N . We determine the identity component of H . For adjoint representations we determine H completely. We also investigate the subgroup GF of GL(V ) preserving a fiber F 6= N of π.