Summary: For an integer constant $d>0$, let ${\text{Γ}}_d$ denote the class of finite groups all of whose non-Abelian composition factors lie in $S_d$; in particular, ${\text{Γ}}_d$ includes all solvable groups. Motivated by applications to graph-isomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, set-stabilizers, group intersections, and centralizers have all been shown to be polynomial-time computable. The most notable gap in the theory has been the question of whether normalizers of subgroups can be found in polynomial time. We resolve this question in the affirmative. Among other new procedures, the algorithm requires instances of subspace stabilizers for certain linear representations and therefore some polynomial-time computation in matrix groups.