Let $G$ be a compact metrizable group and let $H$ be a closed subgroup. Let $d$ be a compatible metric on $G$ that is left-invariant and it is also right invariant with respect to the elements of $H$. That is $$d(g,h)=d(fg,fh)=d(gk,hk),$$ for all $f,g,h\in G$ and $k\in H$.
Then it is possible to well-define a quotient metric $D$ on the homogeneous space $G/H$, which is $G$-invariant (simply by setting $$D(gH,fH):=\min_{h\in H} d(gh,f),$$ for all $g,f\in G$).
My question is about kind of a converse. Suppose that $G$ and $H$ are as above and we are given a compatible $G$-invariant metric $D$ on the homogeneous space $G/H$. Is $D$ a quotient metric of some left-invariant metric $d$ on $G$? I am rather sure that not in general, but I do not know how to look for a counterexample.