Neumann–Praeger covering-subgroup conjecture

Let $A$ be a finite group with a normal subgroup $H$. A subgroup $U$ of $H$ is an $A$-covering subgroup of $H$ if

$$H=\bigcup_{a\in A}U^a.$$

Neumann–Praeger conjecture. Is there a function $f:\mathbb{N}\to\mathbb{N}$ such that whenever $U<H\leq A$, where $H$ is a normal subgroup of the finite group $A$ of index $n$ and $U$ is an $A$-covering subgroup of $H$, one has

$$|H:U|\leq f(n)?$$

This conjecture is important for applications to Kronecker classes of field extensions and concerns whether the index of an $A$-covering subgroup can be bounded in terms of the index of the normal subgroup it covers. It is stated as an open problem of Neumann and Praeger.