  For $k, \ell \in \mathbb{N}$, we introduce the concepts of
$k$-ultrahomogeneity and $\ell$-tuple regularity for finite groups. Inspired by
analogous concepts in graph theory, these form a natural generalization of
homogeneity, which was studied by Cherlin and Felgner and Li as well as
automorphism transitivity, which was investigated by Zhang. Additionally, these
groups have an interesting algorithmic interpretation. We classify the
$k$-ultrahomogeneous and $\ell$-tuple regular finite groups for $k, \ell \geq
2$. In particular, we show that every 2-tuple regular finite group is
ultrahomogeneous.
