Measuring the symmetry of a set in $\mathbb{R}^d$

Question

Suppose is given a finite set of positions in $\mathbb{R}^d$. And a finite group $G$. Is there any way to measure the symmetry of the set related to G? For example if the set consist of vertices of a square it is clear that is symmetric to the dihedral group of eight elements. While if the set has vertices of a rhombus is partially symmetric because there are some elements of $G$ that preserve the positions. And a scalene triangle do not have any symmetry for this group. There is some way to measure that? I am interested at any reference on this.