Large normal subgroup growth and large characteristic subgroup growth

The maximal normal subgroup growth type of a finitely generated group is nlogn. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let Γ be a group and Δ a subgroup of finite index. Suppose Δ has normal subgroup growth of type nlogn, does Γ has normal subgroup growth of type nlogn? We give a positive answer in some cases, generalizing a result of M\"uller and the second author and a result of Gerdau. For instance, suppose G is a profinite group and H an open subgroup of G. We show that if H is a generalized Golod-Shafarevich group, then G has normal subgroup growth of type of nlogn. We also use our methods to show that one can find a group with characteristic subgroup growth of type n^logn.