Let $F$ be an algebraically closed field of characteristic $2$.
I am wondering if the only extensions of $F(t)$, isomorphic to $F(t)$ over $F$ are given by the Frobenius $F(\sqrt t) / F(t)$ and the separable extension $F(t)[x]/(x^2+tx+t^2+t)/F(t)$ up to isomorphism (in the sense of allowing isomorphism both of the source and target field).
In characteristic $0$ this holds by Hurwitz theory (while the case of geometric Frobenius cannot exist), but for example $\mathbb{Q}$ has a lot of non-isomorphic extensions of degree $2$, so in the answer there must be at least some properties of $F(t)$ be used.