No codeword in the Golay code $\mathcal C$ has Hamming weight $1$:
in fact all the weights are even.
Thus, no row of the
generator matrix $\mathcal G$ can have a single $1$ in it, and puncturing
the code by deleting the $i$-th coordinate does not produce any all-zeroes
rows in $\mathcal G$ that might need to be discarded. The punctured code
is thus a $[23, 12, 7]$ code $\mathcal C^*$.
Now, extend the punctured code $\mathcal C^*$by putting the
overall parity check
between the $(i-1)$-th and $i$-th coordinates instead of at the end
as in your description of extension. Remember that the
the $(i-1)$-th coordinate in $\mathcal C^*$ is the same as
the $(i-1)$-th coordinate in $\mathcal C$
while the $i$-th coordinate in $\mathcal C^*$ is the
$(1+1)$-th coordinate in $\mathcal C$
that got moved leftward by one position when the puncturing occurred.
An overall parity check bit is $0$ if the codeword being extended
has even weight and is $1$ if the codeword being extended has odd weight.
(If you do not know this already, you can deduce it as
follows. (i) After extension, each row of the new generator
matrix is an even-weight codeword of the extended code,
and so all the codewords
of the extended code (being sums of rows of the generator
matrix) have even weight, (ii) If a codeword
in the original code had even weight, the extension bit
must necessarily be a $0$ while if the codeword had odd weight
the extension bit must necessarily be a $1$ to satisfy the
condition that all codewords in the extended code have even weight.)
Now, suppose $c \in \mathcal C$ got punctured to $c^* \in \mathcal C^*$ by
deletion of a $0$ bit. Then, $\text{wt}(c^*) = \text{wt}(c)\equiv 0 \mod 2$,
and so the overall parity check being inserted will be a $0$.
On the other hand, if a $1$ bit was deleted in the puncturing
process, then $\text{wt}(c^*) = (\text{wt}(c) - 1)\equiv 1 \mod 2$
and so the overall parity check bit that will be inserted is a $1$.
Thus, the insertion of an overall parity check bit to extend
$\mathcal C^*$ simply puts back into each $c^*$ the bit that was
deleted during the puncturing process to get the punctured codeword $c^*$.
Exercise: Figure out where in the above proof we used
any specific property of Golay codes.
