  We prove automorphy lifting theorems for 2-dimensional Galois representations
of absolute Galois groups of totally real fields when the residual
representation is of "exceptional" type. This exceptional case is when we are
in characteristic 5, the residual representation has projective image
PGL_2(F_5), and the fixed field of its kernel contains a primitive 5th root of
unity. In this case the Taylor-Wiles patching method does not suffice and we
prove our theorem by combining patching with the technique of automorphy by
p-adic approximation.
