  This article is the second part of a series of three articles about
compatible systems of symplectic Galois representations and applications to the
inverse Galois problem. This part is concerned with symplectic Galois
representations having a huge residual image, by which we mean that a
symplectic group of full dimension over the prime field is contained up to
conjugation. A key ingredient is a classification of symplectic representations
whose image contains a nontrivial transvection: these fall into three very
simply describable classes, the reducible ones, the induced ones and those with
huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give
simple conditions under which a symplectic Galois representation with
coefficients in a finite field has a huge image. Finally, we combine this
classification result with the main result of the first part to obtain a
strenghtened application to the inverse Galois problem.

Number Theory