 Solution

Take $T_f$ to consist of $T$ together with the following sentences.

This includes $f(c)=c$ for each constant symbol $c$.

The first pair of axioms makes the interpretation $\vartheta$ a bijection. The remaining schemes say exactly that it preserves every operation and relation, so $(M,\vartheta )\models T_f$ exactly when $M\models T$ and $\vartheta \in \mathrm{Aut}(M)$.

Finally, $T$ is consistent and hence has a model $M$ by part (b). Expanding $M$ by interpreting $f$ as the identity automorphism gives a model of $T_f$, so $T_f$ is consistent.