  Lunardon and Polverino introduced in 2001 a new family of maximum scattered
linear sets in $\mathrm{PG}(1,q^n)$ to construct linear minimal R\'edei
blocking sets. This family has been extended first by Lavrauw, Marino,
Trombetti and Polverino in 2015 and then by Sheekey in 2016 in two different
contexts (semifields and rank metric codes). These linear sets are called
Lunardon-Polverino linear sets and this paper aims to determine their
automorphism groups, to solve the equivalence issue among Lunardon-Polverino
linear sets and to establish the number of inequivalent linear sets of this
family. We then elaborate on this number, providing explicit bounds and
determining its asymptotics.
