Question 1. Why the homomorphism $\varphi_1$ and $\varphi_2$ give rise to isomorphic semidirect product?
From what i see, $\varphi_2=\varphi_1^2$ then $(V\rtimes_{\varphi_1} T)\simeq (V\rtimes_{\varphi_1^2} T)$
What is the explicit isomorphism in this case? Well, I can't see how to relate $\varphi_1$ and $\varphi_2$ with said isomorphism.
Question 2. Why in this case the only non-abelian group is $A_4$?
I know that the semi-direct product is non-abelian when the homomorphism is non-trivial, but why in this case must the group be isomorphic to $A_4$?
Question 3. Finally in the last part, I can't see how Dummit finds the homomorphisms that are mentioned. I have the following (written not rigorously):
$V=\langle a\rangle\times \langle b\rangle$, $T=\langle \lambda\rangle$
I need find the homomorphism $\varphi:V\to \text{aut}(T)$
$|V|=4$ then $|\ker(\varphi)|\in \left\{1,2,4\right\}$ then $\ker(\varphi)=\left\{1,\langle a\rangle, \langle b\rangle, \langle ab\rangle\right\}$
(I) If $\ker(\varphi)=\langle ab\rangle$ then $\varphi(ab)=\lambda^2$ (because $\lambda^2$ is the identity in $\mathbb{Z}_2.$
Now, $\varphi(ab)=\varphi(a)\varphi(b)$ (homomorphism) then $\varphi(a)\varphi(b)=\lambda^2$ implies that $\varphi(a)=\varphi(b)=\lambda$ (obtaining the homomorphism mentioned by dummit in a not so rigorous way)
Therefore, $\varphi_1(a)=\lambda$ and $\varphi_1(b)=\lambda$ has kernel $\langle ab\rangle.$
Why ab centralizes $T$? I have the following: If $T=\langle y\rangle$, $y^3=1$
$(ab)\cdot y=\varphi(ab)(y)=\lambda^2(y)=y$ (because $\lambda^2=Id$)
then $ab y (ab)^{-1}=y$ i.e. $aby=yab $therefore $ab$ centralizes $T$. This is right?
(II) If $\varphi_2$ and $\varphi_3$ have kernels $\langle a\rangle$, $\langle b\rangle$ repectively. Why the three semidirect products are all isomorphics to $S_3\times \mathbb{Z}_{2}$?
Again I cannot see the explicit isomorphism between the semi-direct products.