Let $N$ is normal subgroup of $A$, is $N \rtimes e$ normal subgroup of $G= A \rtimes B$?

Question

Let $G= A \rtimes B$ where $ A,B \leq G$. ( where $ \rtimes $ denotes semi-direct product)

Question- Let $N$ is normal subgroup of $A$, is $N \rtimes e$ normal subgroup of $G= A \rtimes B$ ?

If yes, can we say that $(A \rtimes B)/(N\rtimes e) \cong A/N \rtimes B/e$?

Answer 1

Not necessarily. Consider $A=C_2\times C_2$, $B=C_2$, and let the action of $B$ be given by exchanging coordinates. Then $C_2\times\{e\}$ is normal in $A$, but $(C_2\times \{e\})\rtimes \{e\}$ is not normal in $A\rtimes B$.

For $N$ to be normal, you need it to be both normal in $A$ and "$B$-invariant", that is, for all $b\in B$, the action of $b$ on $A$ has $N^b=N$. In that case, it is easy to verify that the quotient of $A\rtimes B$ by $N\rtimes\{e\}$ is indeed $(A/N)\rtimes B$, with the induced action, and I will let you prove it for yourself.