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$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\W{W}$Let $k$ be a finite field, and let $\W_n(k)$ be the degree $n$ Witt vectors over $k$ (so $\W_1(k) = k$).

Does there exist a subgroup $H \le \SL_2(\W_2(k))$ that maps isomorphically onto $\SL_2(k)$?

If $k$ has characteristic $p \ge 5$, no such subgroup exists. This follows from a well-known argument of Serre (in his "Abelian $\ell$-adic representations" book) which proves slightly more – any subgroup of $\SL_2(\W_2(k))$ which surjects onto $\SL_2(k)$ must be the whole group. It's also easy to check the claim by hand for $k = \mathbf{F}_2$, although Serre's stronger assertion does not work for $\mathbf{F}_2$.

On the other hand, for $k = \mathbf{F}_3$ such a pathological subgroup really does exist (coming from the 2-dimensional characteristic 0 representation of $\GL_2(\mathbf{F}_3)$ that is a vital input in the proof of Fermat's Last Theorem).

What happens for larger fields of characteristic 2 or 3?

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I found the answer myself with the help of a very useful hint from user "nobody" in the comments, so I'm going to post a community-wiki answer in case anyone else finds it useful.

My question is answered by Manoharmayum in this paper:

Manoharmayum, Jayanta, A structure theorem for subgroups of (GL_n) over complete local Noetherian rings with large residual image., Proc. Am. Math. Soc. 143, No. 7, 2743-2758 (2015). ZBL1338.20047.

Lemma 3.7 of the paper confirms that $SL_2(k)$ lifts to a subgroup of $SL_2(W_2(k))$ if and only if $k = \mathbf{F}_3$.

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