Let $N\rtimes Q$ be a semidirect product of groups with normal subgroup $N$ and quotient $Q$.
Let $H\le N$ be a fixed subgroup of finite index $d$. Is it possible to classify the subgroups $U$ of $N\rtimes Q$ with index $d$ such that $U\cap N = H$?
By classify, I would like to at least know that if such a $U$ exists, must it be unique? and if it is not unique, then is the set of all $U$'s satisfying the condition a torsor under some (cohomology?) group?
If this is difficult in general, can we at least do it if we assume $d = 2$?