Isomorphisms preserving invariants

Let $V$ and $W$ be finite dimensional real vector spaces and let $G\subset \text{GL}(V)$ and $H\subset \text{GL}(W)$ be finite subgroups. Assume for simplicity that the actions contain no reflections. Let $Y$ and $Z$ denote the real algebraic varieties corresponding to $\mathbb{R}[V{]}^G$ and $\mathbb{R}[W{]}^H$, respectively. If $V$ and $W$ are quasi-isomorphic, i.e., if there is a linear isomorphism $L\text{\hspace{0.17em}⁣}:V\to W$ such that $L$ sends $G$-orbits to $H$-orbits and $L\text{inv}$ sends $H$-orbits to $G$-orbits, then $L$ induces an isomorphism of $Y$ and $Z$. Conversely, suppose that $f\text{\hspace{0.17em}⁣}:Y\to Z$ is a germ of a diffeomorphism sending the origin of $Y$ to the origin of $Z$. Then we show that $V$ and $W$ are quasi-isomorphic, This result is closely related to a theorem of Strub \cite{Strub}, for which we give a new proof. We also give a new proof of a result of \cite{KrieglLosikMichor03} on lifting of biholomorphisms of quotient spaces.