From local to global similarity of matrix groups

Let G and H be groups of complex n×n matrices. We say that G is an H-like group if every matrix in G is similar to a matrix from H. For several groups H we consider two questions:(A)Is every H-like group (simultaneously) similar to a subgroup of H?(B)Is H the only H-like group containing H? Among other results we prove that the symmetric group Sn is the only Sn-like group containing Sn.