I'm trying to find the normalizer of the Pauli group $G_n$ (as a subgroup of $SU(2^n)$) utilizing Lie algebras, as is done in a reference to find the normalizer of the Heisenberg group $HW(n)$. There, they see its Lie algebra $\mathfrak{hw}(n)$ (the Heisenberg algebra) to figure out a (Lie) algebra containing it as an ideal, that is, the normalizer $N[\mathfrak{hw}(n)]$. The result is the semidirect sum of the Heisenberg algebra and the real symplectic algebra $\mathfrak{sp}(2n,\mathbb{R})$. Then, the corresponding Lie group (called the Clifford group $C_n$ in quantum information community) amounts to the sought-after normalizer (of the Heisenberg group).
In applying the same line of techniques to the Pauli group, I bump into the fact that the Pauli group is not a continuous group in the first place (though, it might be a discrete subgroup of $SU(2^n)$).
Q.) Do we have a Lie-algebra-like things for the Pauli group? More generally, do we have a linear space or algebra (such as Lie algebras) to understand a finite group that enables similar techniques mentioned to find its normalizer? As a side-question, what is ordinary method to find the normalizer of a group in general?