On the base size of a finite group on its action on the lattice of subgroups

Given a finite group $R$, we investigate the base size of the action of the automorphism group of $R$ on the lattice of subgroups of $R$. Our main result shows that this base size is $1$ if and only if $R$ is cyclic. Our motivation arises from a conjecture of Babai on the problem of representing groups as automorphism groups of lattices with a bounded number of orbits.