every small extension $0\to I\to A'\to A\to 0$ of Artin ${{\mathbb C}}$-algebras and every deformation $\xi\co{{\mathscr C}}\subset X_A$ over $A$ of the embedding $C\subset X$ (where $X_A=X\times {\operatorname{spec}}(A)$) an element $\omega(\xi,A')\in{\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)\otimes I$ which is the obstruction to lifting $\xi$ to $A'$ (cf. Definition 5.1 in [@TV]).

The group $G$ acts on deformations of the embedding $C\subset X$ by changing the identification of the central fiber. Let us assume that $0\to I\to A'\to A\to 0$ is a tiny extension, i.e. that $I\cong{{\mathbb C}}$. Then, from the canonicity of minimal obstruction spaces (cf. Exercise 5.8 in [@TV]), it follows that $\omega(\alpha\cdot\xi,A')=\alpha\cdot\omega(\xi,A')$, for all $\alpha\in G$. Therefore, obstructions arising from $G$-equivariant lifting problems are contained in ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G$.

We have to show that all elements of ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G$ are obtained as obstructions to $G$-equivariant lifting problems. For this, it is enough to observe the following. Every element of ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)$ is of the form $\omega(\xi,A')$, for some tiny extension $0\to I\to A'\to A\to 0$ and some deformation $\xi\co{{\mathscr C}}\subset X_A$ over $A$ of the embedding $C\subset X$. By the construction appearing in the proof of Proposition 5.6 in [@TV], the sum of the elements appearing in the $G$-orbit 