Studying finite groups from a combinatorial perspective leads to a demand for combinatorial symmetry and regularity measures. In this context, we develop the concepts of $k$-ultrahomogeneity and $k$-tuple regularity for finite groups. They are inspired by analogous concepts in graph theory and form natural generalizations of previously known group-theoretic properties such as ultrahomogeneity and automorphism transitivity. In my talk, I will discuss both notions and present a classification result.