An (orthogonal) linear representation $\rho:G\to\mathrm O(\Bbb R^d)$ of a finite group is a homomorphism of $G$ into the orthogonal group. I am interested in the resulting matrix group $\mathcal M:=\mathrm{im}(\rho)$. Sometimes, distinct representations of $G$ give me the same matrix group.
Example. Consider the cyclic group $C_5=\langle \sigma\rangle$ generated by a single element $\sigma$ of order five. All representations $\rho$ that map
$$\sigma\mapsto R_{2\pi k/5}\quad \text{(the clock-wise rotation of the plane by $2\pi k/5$ around the origin)}$$
for $k\in\{1,...,4\}$ give me the same matrix group, despite some are considered different representations.
Is there some theory that tells me how many essentially distinct (irreducible) matrix groups I can obtain? Let's say that two matrix groups are essentially the same if they are related by conjugation w.r.t. $\mathrm O(\Bbb R^d)$. I would expect something similar to counting the conjugacy classes of $G$ for obtaining the number of essentially different representations of $G$.
Equivalently: how many essentially distinct irreducible matrix groups are there, that are isomorphic to $G$?