Suppose a finite group $G$ acting on a set $D$ is given. I'm interested in finding a subgroup of $G$ that acts on a subset of $D$ such that $G$ has a maximally large block system (maximal in terms of number of blocks)
E.g., if $D=\{1..10\}$ and $G=\{(1, 4, 7)(2, 5, 8) (3, 6, 9), (1, 7) (2, 8) (3, 9), (9, 10)\}$ A good answer would be the subgroup
$G'={(1, 4, 7)(2, 5, 8) (3, 6, 9), (1, 7) (2, 8) (3, 9)}$ which has as blocks $\{1,2,3\}, \{4,5,6\}$ and $\{7,8,9\}$.