Permutation groups arising from pattern involvement

For an arbitrary finite permutation group $G$, subgroup of the symmetric group $S_{\mathrm{\ell }}$, we determine the permutations involving only members of $G$ as $\mathrm{\ell }$-patterns, i.e., avoiding all patterns in the set $S_{\mathrm{\ell }}\setminus G$. The set of all $n$-permutations with this property constitutes again a permutation group. We consequently refine and strengthen the classification of sets of permutations closed under pattern involvement and composition that is due to Atkinson and Beals.