On the base size of a finite group in 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.