  In this article new cases of the Inverse Galois Problem are established. The
main result is that for a fixed integer n, there is a positive density set of
primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite
extension of the rational numbers. These groups are obtained as projective
images of residual modular Galois representations. Moreover, families of
modular forms are constructed such that the images of all their residual Galois
representations are as large as a priori possible. Both results essentially use
Khare's and Wintenberger's notion of good-dihedral primes. Particular care is
taken in order to exclude nontrivial inner twists.
