Classify the groups of order $20$

Question

Classify the groups of order $20$ (there are five isomorphism types).

$\newcommand{\semi}{\rtimes}\newcommand{\aut}{\operatorname{Aut}}$ Proof. If $G$ is abelian, $G\simeq\Bbb Z/20\simeq\Bbb Z/4\times\Bbb Z/5$ or $\Bbb Z/2\times\Bbb Z/2\times\Bbb Z/5$. Suppose $G$ is not abelian. Then by Sylow theorem, $n_5 =1$ so a Sylow $5$-subgroup $H<G$ is unique. Let $K$ be a Sylow $2$ subgrop of $G$. Then $G\simeq H\semi K$. Note that $H\simeq\Bbb Z/5$. Since $G$ is not abelian, the induced homomorphism $\varphi:K\to\aut(H)\simeq \Bbb Z/4$ is nontrivial. If $K \simeq\Bbb Z/4$ then $\varphi:\Bbb Z/4\to\Bbb Z/4$ by $[1]\mapsto [1]$ or $[2]$ or $[3]$. First and last map are equivalent. Hence there are two non abelian group $H\semi_\varphi K$ in this case. If $K\simeq\Bbb Z/2\times\Bbb Z/2$ then $\varphi:\Bbb Z/2\times\Bbb Z/2\to\Bbb Z/4$ by $([1],[0])\mapsto [0]$ or $[2]$ and $([0],[1])\mapsto [0]$ or $[2]$ are all possible maps. Hence, there are another two non abelian group.

In the proof, I concluded there are total $6$ groups of order $20$. I guess there is a duplication but I can't find. Could you help?

You're extra group is in the $K\simeq\mathbb{Z}/2\times\mathbb{Z}/2$ case, but your choices of $\varphi$ are not clear. There should be $4$ - one leads to an abelian group, the others are equivalent (you should show this) — Robert Chamberlain, Sep 01 '21 at 07:11
@RobertChamberlain I edit the post. Oh there's a trivial map! — one potato two potato, Sep 01 '21 at 07:18
Your last two non-abelian groups differ only by the numbering of the generators of $K$. So they should be isomorphic. — Brauer Suzuki, Sep 01 '21 at 08:56
"but I can't find it" - I find so far this post, this one, or this one. Well, also this one and so on. — Dietrich Burde, Sep 01 '21 at 10:09
@DietrichBurde The meaning of "duplication" I intended was for a group. There are total $5$ isomorphism type but I found $6$. So there must be a 'duplication' among the groups I found. — one potato two potato, Sep 01 '21 at 11:37
I am sorry. I thought you were looking for a reference here. But anyway, you have an answer now. You can find more details in the above posts. Indeed, there are exactly two non-abelian groups of order $20$ up to isomorphism. This is written down in all detail in the literature. — Dietrich Burde, Sep 01 '21 at 11:42

Classify the groups of order $20$

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