# A question about classifying a semidirect product of groups

I was trying to classify all groups of the form $\mathbb Z_{21} \rtimes_{\alpha} \mathbb Z_2$ and show that these groups are $\mathbb Z_{42}$, $D_{42}$, $D_6\times \mathbb Z_7$, and $\mathbb Z_3\times D_{14}$.

I said that if we have $\mathbb Z_{21} \rtimes_{\alpha} \mathbb Z_2$, then we have a homomorphism $\alpha: \mathbb Z_2 \rightarrow \operatorname{Aut}(Z_{21})$. But $\operatorname{Aut}(\mathbb Z_{21}) \cong \mathbb Z^{\times}_{21}$. But we know that $\mathbb Z^{\times}_{21}$ is either isomorphic to $\mathbb Z_{12}$ or $\mathbb Z_2\times \mathbb Z_6$.

Case 1:

We look at $\mathbb Z_2\times \mathbb Z_6$, and so we have the homomorphism $\alpha: \mathbb Z_2 \rightarrow \mathbb Z_2\times \mathbb Z_6$. Let $\mathbb Z_2 = <a>$, $\mathbb Z_2=<a>$, and $\mathbb Z_6=<b>$.

So the possibilities for the homomorphisms are:

a) $\alpha(a) = (e,e)$

b) $\alpha(a)= (a,e)$

c) $\alpha(a) = (a,b^3)$

But know I'm stuck. I'm not sure how I would classify groups from these homomorphisms. In other words, how do I know which homomorphism corresponds to which group? I looked at some examples in the internet, but they did not show the step that I'm looking for. They just gave the possible homomorphisms and then skiped that step. So I'm not sure how I can approach this.