Let $K$ be an imaginary quadratic field and suppose that $E/K$ has complex multiplication by $\mathcal{O}_K$. Let $\psi$ be the Hecke character associated with $E$ and $\mathfrak{f}$ its conductor (i.e it is the integral ideal of $\mathcal{O}_K$ coinciding exactly with the primes of bad reduction of $E$). Let $\mathfrak{a} \triangleleft \mathcal{O}_K$ and $K(\mathfrak{a})$ denote the ray class field of $K$ modulo $\mathfrak{a}$.

Given that $\mathfrak{a}$ is prime to $6\mathfrak{f}$ and $E[\mathfrak{af}] \subseteq E(K(\mathfrak{af}))$ and that $\mathrm{Gal}(K(E[\mathfrak{a}])/K) \hookrightarrow (\mathcal{O}_K/\mathfrak{a})^\times$, show that in fact $\mathrm{Gal}(K(E[\mathfrak{a}]/K) \cong (\mathcal{O}_K/\mathfrak{a})^\times$.

I have been struggling to prove this for a long while now. It is effectively the content of Corollary 5.20 of Rubin's paper found here . I don't particularly understand the assertions regarding the kernel and isomorphism that follow the first paragraph in the proof. Does anyone have any ideas? Help with the other assertions of Corollary 5.20 (iii - v) would also be greatly appreciated.

Cheers!

P.S I have crossposted this from MathSE as I have yet to receive an answer there.