For a field $K$ of characteristics $p>0$, when is a finite purely inseparable extension $F/K$ (with $[F:K]=p^n>1$) such that $F\cong K$?

Question

An example that illustrates the question is: $F=\mathbb{F}_p(t)$ and $K=\mathbb{F}_p(t^p)$, for which $F\cong K$ by Luroth's theorem.

Also, for $p>3$, consider $F=\overline{\mathbb{F}}_p(x,y)$, where $y^3=x^2+1$. If $K=\overline{\mathbb{F}}_p(x^p,y^p)$, then we have $[F:K]=p$ and $F\cong K$.

I suspect that $F\cong K$ is not true in general, and I'm looking for necessary and sufficient conditions for that to hold. Any idea/reference is welcome!