I'm hoping for some ideas/pointers here. I'm experimenting with a Livschitz theorem for functions on a locally compact Abelian group, where the periodic orbit sums take values in a closed subgroup.

If $G$ is a locally compact Abelian group (second countable) and $H$ is a closed subgroup then $G$ and $H$ inherit a natural translation-invariant metric. You can use this to induce a translation-invariant metric on $G/H$.

Does there exist a neighbourhood $N$ of the identity in $G/H$ and a continuous map $f\colon G/H\to G$ such that $x\in f(x)+H$ for all $x\in N$?

EDITED to incorporate simplifying suggestion of Emil Jerabek.