I've run across something that surprises me, so I'm wondering (1) Is it true? and (2) Is it well known? (And if the answers are affirmative, why didn't I know this already?)

Let $G$ be a compact Lie group and let $H$ and $K$ be closed subgroups. So that the question isn't trivial, you can assume that $H$ is subconjugate to $K$. Then we can consider the $WH = NH/H$ space $(G/K)^H$. The claim is that this is a disjoint union of orbits of $WH$.

The first proof I found depends on the Mongtomery-Zippin theorem via a consequence that appears as Lemma 1.1 in Peter May's "Equivariant Orientations and Thom Isomorphisms": If $j\colon \alpha\to\beta$ is a homotopy between $G$-maps $G/H\to G/K$, then $j$ factors as the composite of $\alpha$ and a homotopy $c\colon G/H\times I \to G/H$, where $c(eH,t) = c_tH$ for a path $c_t$ in the centralizer $C_GH$ of $H$, starting at the identity. Reinterpreting in terms of $(G/K)^H$, this implies that, if $\alpha$ and $\beta$ are two points in the same path component of $(G/K)^H$, then $\beta = c\alpha$ for some $c$ in the identity component of $C_GH$. In particular, they are in the same $WH$-orbit. Hence, each $WH$-orbit consists of a union of path components of $(G/K)^H$.

I think another proof could be found by looking at the tangent plane at an $H$-fixed point in $G/K$, as an $H$-representation, and observing that the only $H$-trivial directions are those in the direction of the $NH$-action. (But I haven't worked this out completely.)

So, is this true? Is it already well known? If it is true, it has some interesting implications for equivariant ordinary homology.