contain $\mathcal V$ in a strict sense, since, at points of $V\backslash{\mathcal V}$, the conjugacy class in $G$ of the isotropy subgroup of Stab$(H,G)$ could be smaller than the conjugacy class in $G$ of the isotropy subgroup of $G$.

Now, $H$ is an invariant subgroup of Stab$(H,G)$ and a subgroup of all the isotropy subgroups of (Stab$(H,G),V)$. Thus, the action of Stab$(H,G)$ in $V$ defines a linear group $K$ (and the group space $(K,V)$), isomorphic to the quotient group Stab($H$, $G$)/$H$, through the relation

[$${(sH)\cdot v=s\cdot v,\qquad s\in {\rm Stab}(H,G),\ sH\in
{\rm Stab}(H,G)/H.}\label{K}$$]{} So, we can conclude that [*the intersection of a $G$-orbit of $S$ with ${\mathcal V}$ is an orbit of $(K,V)$*]{}. Obviously, the orbit spaces $V/K$ and $V/{\rm
Stab}(H,G)$ are isomorphic and can be identified.

We can rephrase the result just obtained, by claiming that the points of $\widehat S$ are in a one-to-one correspondence with the points of a, possibly proper, subset $\widehat{\mathcal V}$ of the principal stratum $\widehat\Sigma$ of the orbit space $V/K$ and the following relations hold true:

[$${\overline{\widehat S}\,=\,p(V)\,\supset\,
p(\Sigma)\,\supseteq\, p({\mathcal V})\, =\, \widehat S.}\label{pS}$$]{} Since, as stressed, we know how to parametrize a principal stratum, our problem is reduced to the determination of 