Étale triviality of finite equivariant vector bundles

Let H be a complex Lie group acting holomorphically on a complex analytic space X such that the restriction to Xred of every H-invariant regular function on X is constant. We prove that an H-equivariant holomorphic vector bundle E over X is H-finite, meaning f1(E) = f2(E) as H-equivariant bundles for two distinct polynomials f1 and f2 whose coefficients are nonnegative integers, if and only if the pullback of E along some H-equivariant finite étale covering of X is trivial as an H-equivariant bundle.