For generalized Artin-Schreier extensions over F_{p^4}, a subextension descends to F_p or F_{p^2} exactly when its Galois subgroup is stable under the corresponding Frobenius, with a full classification in the bi-cyclic case.
Goss, Basic structures of function fields arithmetic
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On the Weil descent of Artin-Schreier algebraic function fields over finite fields
For generalized Artin-Schreier extensions over F_{p^4}, a subextension descends to F_p or F_{p^2} exactly when its Galois subgroup is stable under the corresponding Frobenius, with a full classification in the bi-cyclic case.