Linear set in projective spaces over finite fields plays central roles in the study of blocking sets, semifields, rank-metric codes and etc. A linear set with the largest possible cardinality and the maximum rank is called maximum scattered. Despite two decades of study, there are only a few number of known maximum scattered linear sets in projective lines, including the family constructed by Csajbók, Marino, Polverino and Zanella 2018, and the family constructed by Csajbók, Marino, Zullo 2018 (also Marino, Montanucci, and Zullo 2020). This paper aims to solve the equivalence problem of the linear sets in each of these families and to determine their automorphism groups.