I'm trying to understand the proof of Lemma 1 of Swinnerton-Dyer, On $\ell$-adic representations and congruences for coefficients of modular forms.
The aim is to show that if $G$ is a closed subgroup of $\mathrm{GL}_2(\mathbb Z_\ell)$ ($\ell>3$) whose image under reduction mod $\ell$ contains $\mathrm{SL}_2(\mathbb F_\ell)$, then $G$ contains $\mathrm{SL}_2(\mathbb Z_\ell)$.
Since $G$ is closed, it is enough to show for each $n>0$ that $G_n\supset\mathrm{SL}_2(\mathbb Z/\ell^n\mathbb Z)$, where $G_n$ is the image of $G$ in $\mathrm{GL}_2(\mathbb Z/\ell^n\mathbb Z)$. This holds by hypothesis for $n=1$.
Let $H_n$ be the kernel of the map $$\mathrm{SL}_2(\mathbb Z/\ell^n\mathbb Z)\to\mathrm{SL}_2(\mathbb Z/\ell^{n-1}\mathbb Z).$$
The proof claims that the result will follow by induction if we can prove that for each $n>1$, $H_n\subset G_n.$ I am struggling to understand why this is the case.
We have the following diagram, where the rows are exact.
The right downard arrow is an isomorphism by induction. The left downward arrow is an isomorphism if we assume that $H_n\subset G_n$. If the top-right map were surjective, then the result would follow by the five-lemma.
But a priori, I don't see why this should be the case, since the image of the intersection of two groups is not necessarily the intersection of the images.
Is there something obvious that I'm missing?