Reduced designs constructed by Key-Moori Method 2 and their connection with Method 3

For a 1-(v,k,λ) design D containing a point x, we study the set Ix, the intersection of all blocks of D containing x. We use the set Ix together with the Key-Moori Method 2 to construct reduced designs invariant under some families of finite simple groups. We also show that there is a connection between reduced designs constructed by Method 2 and the new Moori Method 3.