= Solution
A model of $T_G$ consists of a model $M\models T$ and automorphisms
$$
\vartheta_g=(f_g)^M\in\operatorname{Aut}(M)
$$
such that $\vartheta_g\vartheta_h=\vartheta_{gh}$ and $\vartheta_g\ne\vartheta_h$ whenever $g\ne h$. Thus $g\mapsto\vartheta_g$ is an injective homomorphism $G\hookrightarrow\operatorname{Aut}(M)$. Conversely, any such embedding supplies interpretations of the $f_g$ and hence a model of $T_G$.
Therefore $G$ is $T$-good exactly when $T_G$ has a model. By part (b), this is equivalent to consistency of $T_G$, and hence
$$
\boxed{G\text{ is }T\text{-bad}\quad\Longleftrightarrow\quad T_G\text{ is inconsistent}.}
$$
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