Equivariant morphisms to $G/H$ are locally trivial?

Question

Let $X$ be an algebraic variety over $\mathbb{C}$ and $G$ is a complex algebraic group which acts on X. Fix also an algebraic subgroup $H \subset G$ and consider a $G$-equivariant morphism $f\colon X \rightarrow G/H$. Is it true that $f$ is locally trivial in standard analytic topology (or etale)?