Here’s a strategy that sometimes works but is likely hopeless in general. Suppose $G$ acts on a complex $K$ with the property that each finite subgroup of $G$ fixes a point. For example, if the complex $K$ happens to enjoy the CAT(0) property, this will be satisfied.

In this situation, suppose $H$ is a finite subgroup. It has a fixed subspace $X \subset K$. Each $g \in G$ sends the fixed-point set $X$ for $H$ to the fixed-point set for $gHg^{-1}$, so if $g$ normalizes $H$, then $g$ actually preserves $X$, and if $X$ has the property that $H$ is its pointwise stabilizer, the converse holds; namely, if $g$ preserves $X$, then $g$ normalizes $H$.

Anyway, there are standard techniques to attempt to compute a presentation of a group from its action on a simply connected complex, so if $X$ enjoys this property, then you might attempt to compute a presentation for $N_G(H)$ this way.

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