Considering you want the coordinates for a animated character, you can use the bmesh module and the dependency graph to do that.
bmesh will provide mesh information and the dependency graph is used to retreive this information considering the mesh transformations.
From that, we can get access to vertices that correspond to the vertex group you want.
The principle is to use the active deform layer (vertex groups) of the bmesh and use it over the vertices to have vertex information about this layer.
This access provides a map containing the weights per group index. Here we just need to keep vertices that have the information for the group index we want.
Here is a full commented code (but feel free to ask if anything is not clear enough):
import bpy
import bmesh
Get the object
obj = bpy.context.collection.all_objects["Beta_Surface"] #Your object name here
Get the vertex group
vertex_group = obj.vertex_groups["mixamorig:LeftHand"] #Your vertex group name here
And its index
vg_index = vertex_group.index
Get the dependency graph (used to have mesh information with animation applied)
depsgraph = bpy.context.evaluated_depsgraph_get()
Create a bmesh object
bm = bmesh.new()
Get the information considering the dependency graph
bm.from_object( obj, depsgraph )
Get the deform (vertex groups) active layer (assuming there is one here)
deform = bm.verts.layers.deform.active
The object world matrix
world_matrix = obj.matrix_world
Loop over the vertices keeping only the one attached to the group
v[deform] gets vertex group information as (group index, vertex weight) map
for vert in [v for v in bm.verts if vg_index in v[deform]]:
# vertex coordinate in object space
co = vert.co
# vertex coordinate in world space
world_co = world_matrix @ co
print(co, world_co)