the cotangent space to ${{\mathscr M}}^\phi$ at $s$. By assumption $H^0(C, \Omega^{\otimes 2}_C)^G\neq\{ 0\}$. It is well-known (see [@bl], Lemma 2.1) that ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/{{\mathbb P}}},{{\mathscr O}}_C)^G= H^1(C,{{\mathcal N}}_{C/{{\mathbb P}}})^G=0$, so that the $G$-invariant Hilbert scheme ${{\mathscr H}}^G$ is smooth at $[C\subset{{\mathbb P}}]$. In case ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G=0$, then the exact sequence (\[exact\]) shows that the inclusion ${{\mathscr H}}_X^G\subseteq{{\mathscr H}}^G$ is strict.

When ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G\not=0$, we need the interpretation of this space as an obstruction space, as provided by Lemma \[obstruction3\] below. It then tells us that the $G$-equivariant deformation theory of $C$ in the quadric $X$ is obstructed. So the $G$-invariant Hilbert scheme ${{\mathscr H}}_X^G$ is singular at $[C\subset X]$ and then the inclusion ${{\mathscr H}}_X^G\subseteq{{\mathscr H}}^G$ is also strict.

\[obstruction3\]A minimal obstruction space for the $G$-equivariant deformation theory of $C$ in the quadric $X$ is ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G$.

We proved in Theorem 1.1 in [@bl] that ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)$ is a minimal obstruction space for the deformation theory of $C$ in the quadric $X$ and so we only need to check that this retains this interpretation in a $G$-equivariant setting. This is a matter of carefully going through the definitions.

Let $\omega$ be the map which assigns to 