Very similar to David Gao's and YCor's answers: let $\lambda<\kappa$ be infinite cardinals, and let $R$ be the collection of equivalence classes of subsets of $\kappa$ having cardinality at most $\lambda$ determined by the equivalence relation $A\sim B$ iff $|A\triangle B|<\lambda$.  It's easy to prove that for any two sets $A, B\subset \kappa$ having the same cardinality, there is a permutation $\pi$ of $\kappa$ with $\pi(A)=B$, and this $\pi$ induces an automorphism of $R$ taking the equivalence class of $A$ to the equivalence class of $B$.