Embedding problems with bounded ramification over global fields of positive characteristic

Let K/K₀ be a finite Galois extension of global fields of positive characteristic p. We prove that every finite embedding problem with solvable kernel H over K/K₀ is properly solvable if it is weakly locally solvable and the number of the roots of unity in K is relatively prime to |H|. Moreover, the solution can be chosen to coincide with finitely many (given in advance) weak local solutions. Finally, and this is the main point of this work, the number of primes of K₀ that ramify in the solution field is bounded by the number of primes of K₀ that ramify in K plus the number of prime divisors of |H|, counted with multiplicity. This result completes the main theorem of Jarden and Ramiharimanana (Proc. Lond. Math. Soc. 117 (2018) 149–191) that demands that p does not divide |H|.