Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis (e(j))(jgreater than or equal to1). Given an operator T from L-c(infinity) to L-1(X), we consider the vector-valued extension (T) over tilde of T given by (T) over tilde(Sigma(j)f(j)e(j)) = Sigma(j)T(f(j))e(j). We prove a weighted integral inequality for the vector-valued extension of the Hardy-Littlewood maximal operator and a weighted Fefferman-Stein inequality between the vector-valued extensions of the Hardy-Littlewood and the sharp maximal operators, in the context of Orlicz spaces. We give sufficient conditions on the kernel of a singular integral operator to have the boundedness of the vector-valued extension of this operator on L-p (X, Wdmu; E) for 1 < p < infinity and for a weight W in the Muckenhoupt class A(p)(X). Applications to singular integral operators on the unit sphere S-n and on a finite product of local fields K-n are given. The versions of all these results for vector-valued extensions of operators on functions defined on a homogeneous space X and with values in a UMD Banach lattice are also given.16117197