Unicity of the Perfect Golay Binary Code

Let $C_1$ and $C_2$ be two perfect binary codes of length $23$, rank $12$ and minimum distance $7$. As you observed, extending either of them with an overall parity check bit gives you a code that is equivalent to $G_{24}$. Denote these two extended codes $C_1^+$ and $C_2^+$.

Let's fix a copy $G$ of $G_{24}$. We know that $C_1^+$ is equivalent with it, so there exists a coordinate permutation $\alpha\in S_{24}$ such that $\alpha(C_1^+)=G$. We don't know where the extension bit of $C_1^+$ went in this permutation, so let's call its new position $i=\alpha(24)$. Similarly, we see that there exists a permutation of bit positions $\beta\in S_{24}$ such that $\beta(C_2^+)=G$. The other extension bit was mapped to position $j=\beta(24)$.

By transitivity of the automorphism group of $G$ there exists an automorphism $\sigma$ of the code $G$ such that $\sigma(i)=j$. It follows that the composed permutation of bit positions, $\beta^{-1}\sigma\alpha$ takes the extension bit of $C_1^+$ to the extension bit of $C_2^+$, and hence gives the desired equivalence between the codes $C_1$ and $C_2$.

Essentially we get $C_1$ by puncturing the $i$th bit from $G$ and $C_2$ by puncturing the $j$th bit from $G$. The automorphism $\sigma$ then takes one punctured code to the other.