Let G be a reductive group satisfying the Harish-Chandra condition defined over a totally real field F, E/F a finite cyclic extension of fields. With further assumptions on G, by constructing an explicit morphism between eigenvarities, we prove that every p-adic family of p-adic automorphic representations of G over F can be lifted to a family of p-adic automorphic representations of G over E such that , at every classical point, the lifting is just the classical weak base change lifting. The key ingredients in the theory are: (1) a twisted p-adic trace formula for G/E ; (2) a p-adic fundamental lemma and an equation between p-adic trace formula and twisted p-adic trace formula; (3) a second construction of a twisted eigenvariety of G/E.