Let $\ell \ge 5$ be a prime and $G$ be a closed subgroup of $\operatorname{GL}_2 \mathbb{Z}_\ell$ whose image in $\operatorname{GL}_2 \mathbb{F}_\ell$ is $\operatorname{GL}_2 \mathbb{F}_\ell$. Then $G = \operatorname{GL}_2 \mathbb{Z}_\ell$?

In Serre's "Abelian $l$-adic representations and elliptic curves", the author shows its $\operatorname{SL}_2 \mathbb{Z}_\ell$-version. How can I show the highlighted statement from it?

It suffices to show that $H := G \cap \operatorname{SL}_2 \mathbb{Z}_\ell$ surjects onto $\operatorname{SL}_2 \mathbb{F}_\ell$.

EDIT:

In comments YCor pointed that this is false. I ask this question for the following:

Let $E$ be an elliptic curve over a number field $K$, $\ell \ge 5$ a prime. Assume that the mod $\ell$ representation $G_K \to \operatorname{GL}_2 \mathbb{Z}/\ell$ is surjective. Then is the $\ell$-adic representation $G_K \to \operatorname{GL}_2 \mathbb{Z}_\ell$ surjective?

In the proof of lemma 2.4 of González-Jiménez and Najman - An algorithm for determining torsion growth of elliptic curves, the authors say that this is true by the lemma of Serre that I mentioned above.

How can I show this without first highlighted statement?

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