Let $G$ be a Lie group, and $H$ be a closed subgroup of $G$. Consider the homogeneous space $H\backslash G$. I wonder if for any $x \in H\backslash G$, there is a Borel measurable map $\sigma:H\backslash G \to G$ which is continuous in a neighborhood of $x$ and satisfies
$$\pi \circ \sigma = id_{H\backslash G},$$
where $\pi:G\to H \backslash G$ is the natural projection.
Maybe this is true for more general topological groups (LCH and second countable). I also believe this is true if we add some stronger condition on $H$ like if $H$ is a lattice in $G$. But I am not very sure about this statement. I mostly only care about $G:=\text{SL}(d,\mathbb R)$ so please feel free to add connectedness and semisimplicity etc. conditions on $G$.
Please let me know any specific reference if this is a theorem in any book, or if there are any counterexamples.