  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 Csajb\'ok, Marino, Polverino and Zanella 2018, and the family
constructed by Csajb\'ok, 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.
