Considering quantum codes as codes over $F_2$

Let me briefly elaborate on my comment. I'm not an expert on coding theory but I certainly know the symplectic formalism. If you want to know more about codes, I would recommend that you read "Quantum Error Correction Via Codes Over GF(4)" by Calderbank, Rains, Shor, and Sloane for the $mathbb{F}_4$ formulation or "Quantum error-correcting codes and their geometries" by Ball, Centelles, and Huber for a $mathbb F_2$ formulation.

I assume that you know that any element of the $n$-qubit Pauli group $P_n$ can be written uniquely as $W(z,x,t) = i^t Z(z)X(x)$ for $tinmathbb{Z}_4$, $z,xinmathbb{F}_2^{n}$ where $Z$ and $X$ are defined as usual as $Z(z)|urangle = (-1)^{zcdot u}$, $X(x)|urangle = |u+xrangle$. Note that the centre of $P_n$ is $langle i mathbb{I}rangle$. We get a projection: $$ pi : ; P_n rightarrow mathbb{F}_2^{2n}, quad W(z,x,t) mapsto (z,x). $$ Another way of seeing this is that $P_n$ is a group of order $2^{2n+2}$ which is "almost" extraspecial, i.e. the quotient $P_n/Z(P_n)$ is a (discrete symplectic) vector space. It is symplectic since $$ W(v,t)W(u,s) = (-1)^{[v,u]}W(u,s)W(v,s) $$ for $[v,u]:= v_zcdot u_x - v_x cdot u_z$ the standard symplectic product on $mathbb F_2^{2n}$.

Since $pi$ is a homomorphism, subgroups $Ssubset P_n$ project to linear subspaces $bar S:=pi(S)subsetmathbb F_2^{2n}$. By the above commutation relation, $S$ is Abelian if and only if $bar S$ is isotropic, i.e. $Ssubset S^perp$ (self-orthogonal in coding language). Now it should be clear that any $[[n,k]]$ stabiliser code $Q$ given by a stabiliser group $S$ projects to an isotropic subspace $bar S$. Since $S$ has rank $n-k$, the dimension of $bar S$ is $n-k$, too.

A code word $sinbar S$ is of the form $s=(z_1,dots,z_n,x_1,dots,x_n)$. Its weight is defined as $mathrm{wt}(s)=|{(z_i,x_i)neq 0}|$. The automorphism group of $bar S$ is then defined within the symplectic weight-preserving transformations on $mathbb F_2^{2n}$. The latter are exactly the pair-wise coordinate permutations $S_n$ times the "local" symplectic group $mathrm{Sp}_2(mathbb F_2)$ acting on each symplectic pair independently, i.e. it has order $6^n n!$.

Now for the automorphism group of $Q$. Any symplectic map on $mathbb F_2^{2n}$ is induced from a Clifford unitary and any two Cliffords induce the same map if and only if they differ by a Pauli operator. This means that $mathrm{Aut}(bar S)$ is basically $mathrm{Aut}(Q)$ up to multiplication with Pauli operators. More precisely, $mathrm{Aut}(bar S) = mathrm{Aut}(Q)/S$. The only difference between the two comes from the projection $pi$.

Finally, let me remark that by mapping $(1,0) mapsto 1$ and $(0,2)mapsto omega$ we get an additive mapping between $mathbb F_2^{2n}$ and $mathbb F_4^n$ which maps the isotropic linear codes in the above sense to self-orthogonal additive codes over $mathbb F_4$. In this way, stabiliser codes fit better into the framework of coding theory, but this does not change the structure.