Recently, My Professor taught us about group extension. It is the following: A group $G$ is an extension of $Q$ by $N$ if we have the following short exact sequence:$1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$.
After learning this definition I observed that for $n \geq 3$, the symmetric group on $n$ letters $S_{n}$ is an extension of $\mathbb{Z_{2}}$ by the alternating group on $n$ letters $A_{n}$, that is, we have an exact sequence: $1 \rightarrow A_{n} \rightarrow S_{n} \rightarrow \mathbb{Z_{2}} \rightarrow 1$. From here, I came up with the following question that I am completely stuck at.
If $G$ is any nontrivial group, does there exist a group $H$ that is an extension of a group $Q$ by $G$ such that $N \unlhd H$ implies $N \subseteq G$? (Here, in this question, I have assumed that $H \neq G$.)
Please help me.
