Rigidity of homomorphisms of algebraic groups

Let $G$ be a linearly reductive algebraic group and $H$ an algebraic group, both over an algebraically closed field $k$. We will show the existence a parameter space $M$ for homomorphisms from $G$ to $H$. The action of $H$ on itself by conjugation induces an action on $M$, for which we will show that each orbit is open. This gives back a result of Vinberg and Margaux: the set of $H$-orbits in $M$ is unchanged when $k$ is replaced with an algebraically closed field extension.