When does a homogeneous space define a fibration?

Question

Let $G$ be a locally compact and $\sigma$-compact group acting continuously and transitively on locally compact Hausdorff $X$. Then if $x_0 \in X$ and $H_{x_0}$ denotes the isotropy group at $x_0$ we have the homeomorphism $G/H_{x_0} \overset{\phi}{\simeq} X$, $\phi(gH_{x_0}) = gx_0$. Is it true that we always have a fibration $H_{x_0} \hookrightarrow G \to X$? If not, what are sufficient conditions to have a fibration?