  This article is the third and last part of a series of three articles about
compatible systems of symplectic Galois representations and applications to the
inverse Galois problem.
  This part proves the following new result for the inverse Galois problem for
symplectic groups. For any even positive integer n and any positive integer d,
PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occurs as a Galois group over the rational
numbers for a positive density set of primes l.
  The result is obtained by showing the existence of a regular, algebraic,
self-dual, cuspidal automorphic representation of GL_n(A_Q) with local types
chosen so as to obtain a compatible system of Galois representations to which
the results from Part II of this series apply.
