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?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\W{W}$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?