I was reading of the hamming distance on the symmetry group of a finite set $X$. The distance was defined as follows: $$\text{d}^H (\sigma_1 , \sigma_2) = \frac{| { x \in X : \sigma_1(x) \neq \sigma_2(x) }|}{|X|}$$
where $\sigma_1 , \sigma_2 \in \text{Sym}(X)$. This definition gives us a way to measure the distance between two permutations of a finite set $X$. Now I was wondering if this definition can be extended to matrices. In particular, we know that $\phi: \text{Sym}(X) \to \text{GL}_{|X|}(\mathbb{Z}_2)$ is a faithful representation (Hence we can actually conclude that $\phi$ is a group isomorphism since $X$ is finite).
Hence is there a way to define $\text{d}^H (g_1 , g_2)$ where $g_1 , g_2 \in \text{GL}_{|X|}(\mathbb{Z}_2)$ such that this distance coincides with its hamming distance on the symmetry group?