I have some field $Q$ (it is in fact $\mathbb{Q}_p$ but I don't think that it is important here), with two extensions $K_1$ and $K_2$. The field $K_2=Q(\alpha_i)_i$ is the splitting field of some polynomial $H\in Q[X]$ and $K_1=Q(\xi)$ is the rupture field of some polynomial $X^l-p$. We assume that we know the groups $\operatorname{Gal}(K_1K_2/K_1)$ (it is a cyclic group) and $\operatorname{Gal}(K_1/Q)$ (Galois group of the rupture field of $X^l-p$).
Question: is there some easy way to get $\operatorname{Gal}(K_2/Q)$? with some additionaly hypothesis if needed?
In good circumstancies we have $\operatorname{Gal}(K_1K_2/Q)=\operatorname{Gal}(K_1/Q)\times\operatorname{Gal}(K_2/Q)$ and $\operatorname{Gal}(K_1K_2/Q)/\operatorname{Gal}(K_1K_2/K_2)=\operatorname{Gal}(K_2/Q)$ but I don't see how to combine that to get $\operatorname{Gal}(K_2/Q)$.
As pointed out by Arturo Magidin in the comments, knowing $N$ and $G/N$ are not enough to determine $G$ up to isomorphism, even in the simplest case where $N$ and or $G/N$ is cyclic.
Think about the case where $G$ has order $4$ and $N$ has order $2$. Both groups $N$ and $G/N$ are cyclic of order $2$, whatever $G$ is. From this information, you cannot decide whether $G$ is cyclic or isomorphic to the Klein group.
Unless you have precise information on $H$ or $K_2$, it is impossible to give a satisfactory answer to your question.