Reductive subgroups of reductive groups in nonzero characteristics

Let G be a (possibly nonconnected) reductive linear algebraic group over an algebraically closed field k, and let N ? ?. The group G acts on GN by simultaneous conjugation. Let H be a reductive subgroup of G. We prove that if k has nonzero characteristic then the natural map of quotient varieties HN / H ? GN / G is a finite morphism. We use methods introduced by Vinberg, who proved the same result in characteristic zero. As an application, we show that if ? is a finite group then the character variety C(?, G) of closed conjugacy classes of representations from ? to G is finite.