= Solution
Let
$$
K_2=\mathbb F_{p^2}((t)),
\qquad
L=\mathbb F_{p^2}((u)),
\qquad u^{p^2-1}=t.
$$
The extension $K_2/K$ is the unramified quadratic extension. Since $\mathbb F_{p^2}^\times$ contains all $(p^2-1)$st roots of unity, $L/K_2$ is a cyclic, tamely and totally ramified extension of degree $p^2-1$, with automorphisms $u\mapsto\zeta u$.
The Frobenius automorphism of $\mathbb F_{p^2}/\mathbb F_p$ extends by fixing $u$ and conjugates $\zeta$ to $\zeta^p$. Hence $L/K$ is Galois, its inertia group is
$$
G_0(L/K)=\operatorname{Gal}(L/K_2)\cong\mathbb Z/(p^2-1)\mathbb Z,
$$
and its residue-field Galois group is
$$
G_{-1}(L/K)/G_0(L/K)
\cong\operatorname{Gal}(\mathbb F_{p^2}/\mathbb F_p)
\cong\mathbb Z/2\mathbb Z.
$$
The tame ramification also gives $G_1=1$.
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