Linear maps preserving orbits

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Let H⊂(V) be a connected complex reductive group where V is a finite-dimensional complex vector space. Let v∈ V and let G={g∈(V) gHv = Hv}. Following Raïs we say that the orbit Hv is characteristic for H if the identity component of G is H. If H is semisimple, we say that Hv is semi-characteristic for H if the identity component of G is an extension of H by a torus. We classify the H-orbits which are not (semi)-characteristic in many cases.