Localizability of the embedding problem with symplectic kernel

In this paper we consider the question of how much information is supplied by local solutions to a global embedding problem for the special case in which the normal subgroup belonging to the given group extension is the projective symplectic group PSp(2m, q). It is proved that for suitable Galois extensions K of a given number field k (which constitute part of the data of the embedding problem), the local solutions in a sense determine whether or not an extension K ? K, Galois over k, with $\text{G(}\text{L}\text{K}\text{) ≈ PSp(2m, q)}$, represents a global solution to the embedding problem.