Realizability and admissibility under extension of $p$-adic and number fields

A finite group G is K -admissible if there is a G -crossed product K -division algebra. In this manuscript we study the behavior of admissibility under extensions of number fields M/K . We show that in many cases, including Sylow metacyclic and nilpotent groups whose order is prime to the number of roots of unity in M , a K -admissible group G is M -admissible if and only if G satisfies the easily verifiable Liedahl condition over M .