Large normal subgroup growth and large characteristic subgroup growth

The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $\mathrm{\Gamma }$ be a group and $\mathrm{\Delta }$ a subgroup of finite index. Suppose $\mathrm{\Delta }$ has normal subgroup growth of type $n^{\log n}$, does $\mathrm{\Gamma }$ has normal subgroup growth of type $n^{\log n}$? 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 $n^{\log n}$. We also use our methods to show that one can find a group with characteristic subgroup growth of type $n^{\log n}$.