${{\mathbb P}}$ containing the $G$-invariant canonical curve $C$. Then the $G$-invariant Hilbert scheme ${{\mathscr H}}_X^G$ (of $G$-curves contained in $X$) is a proper subscheme of the $G$-invariant Hilbert scheme ${{\mathscr H}}^G$.

In what follows we use the following notational convention. Given an immersion $Y\hookrightarrow Z$ of schemes, we write ${{\mathcal C}}_{Y/Z}$ for its conormal sheaf on $Y$. This is an ${{\mathscr O}}_Y$-module. The normal sheaf ${{\mathcal N}}_{Y/Z}$ of this immersion is by definition the ${{\mathscr O}}_Y$-dual of ${{\mathcal C}}_{Y/Z}$.

In the proof of Lemma 2.2 in [@bl], we observed that to the nested embeddings $C\subset X\subset{{\mathbb P}}$ is associated a short exact sequence $$0\to(\Omega_C^{\otimes 2})^\vee\to{\mathcal C}_{C/{{\mathbb P}}}\to{\mathcal C}_{C/X}\to 0,$$ which, by Lemma 2.2 in [@bl], determines an exact sequence $$0\to H^0(C,{{\mathcal N}}_{C/X})\to H^0(C,{{\mathcal N}}_{C/{{\mathbb P}}})\to H^0(C, \Omega^{\otimes 2}_C)\to {\operatorname{Ext}}_C^1({{\mathcal C}}_{C/X}, {{\mathscr O}}_C)\to 0.$$ Passing to $G$-invariants is an exact functor, and so we get the exact sequence $$\label{exact}
0\to H^0(C,{{\mathcal N}}_{C/X})^G\to H^0(C,{{\mathcal N}}_{C/{{\mathbb P}}})^G\to H^0(C, \Omega^{\otimes 2}_C)^G\to {\operatorname{Ext}}_C^1({{\mathcal C}}_{C/X}, {{\mathscr O}}_C)^G\to 0,$$ where $H^0(C,{{\mathcal N}}_{C/X})^G$ and $H^0(C,{{\mathcal N}}_{C/{{\mathbb P}}})^G$ identify with the tangent space at $[C\subset X]$ to the scheme ${{\mathscr H}}_X^G$ resp.  the tangent space at $[C\subset{{\mathbb P}}]$ to the scheme ${{\mathscr H}}^G$, while $H^0(C, \Omega^{\otimes 2}_C)^G$ identifies with 