For any group H, the wreath product H wr Sym2 of H with the symmetric group on two points is imperfect. In particular, every group can be embedded as a two-step subnormal subgroup of an imperfect group of roughly the same cardinality (2|H|2).
The wreath product of any group H with the symmetric group on two points is incomplete. Consequently, every group can be embedded as a two-step subnormal subgroup in an incomplete group of similar size (2|H|2).