Let $E$ be an elliptic curve with CM by an order $\mathcal O$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to get an isomorphism $\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\mathbb Z/N\mathbb Z) $. Complex multiplication on $E$ induces a homomorphism $(\mathcal O/ N\mathcal O )^\times\rightarrow \operatorname{GL}_2(\mathbb Z/N\mathbb Z)$. Let $C_N$ be the image.

Is $C_N$ equal to its centralizer in $\operatorname{GL}_2(\mathbb Z/N\mathbb Z)$?

This should occur in some cases, and should give rise to a rational point on certain modular curves. See Burcu Baran: Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem, especially Proposition 4.1.