Admissibility and field relations

Let K be a number field. A finite group G is called K-admissible if there exists a G-crossed product K-division algebra. K-admissibility has a necessary condition called K-preadmissibility that is known to be sufficient in many cases. It is a 20-year-old open problem to determine whether two number fields K and L with different degrees over ℚ can have the same admissible groups. We construct infinitely many pairs of number fields (K,L) such that K is a proper subfield of L, and K and L have the same preadmissible groups. This provides evidence for a negative answer to the problem. In particular, it follows from the construction that K and L have the same odd order admissible groups.