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 $G/H$.

Given a continuous function $f\colon X\to G/H\ $ and a point $x\in X$, does there exist a neighbourhood $N$ of $x$ and a continuous function $F\colon X\to G$ such that $f(x)=F(x)+H\ $ for $x\in N$?