I am trying to classify the non-abelian group of order $20$.
We know that $K$ is a subgroup of order $5$ is normal. $H$ is a subgroup of order $4$.
Case $1:$ let $H \cong \mathbb{Z_2} \times \mathbb{Z_2}$
Then we need to find a homomorphism $\varphi:H \to Aut(\mathbb{Z_5})$
We need to find an element of order $2$ in Aut($\mathbb{Z_5}$).
The element of order $2$ in given by $\sigma_4:(1_5) \to 4_5$.
Now possible nontrivial homomorphisms are
$\varphi_1(1_2,0) \mapsto \sigma_4$ and $\varphi_1(0,1_2) \mapsto id$
Another possible homomorphism is
$\varphi_2(1_2,0_2) \mapsto \sigma_4$ and $\varphi_2(0_2,1_2) \mapsto \sigma_4$
Then $K \rtimes_{\varphi_1} H$ and $K \rtimes_{\varphi_2} H$ are two possible groups. Is this ok? Or are the two groups isomorphic?
Case $2:$ let $H \cong \mathbb{Z_4}$
Then we need to find a homomorphism using the following
$\varphi'_1:H \to Aut(K)$
Where $\varphi'_1(1_4) \mapsto \sigma_2$.
Then $\sigma_2:1_5 \to 2_5$ is the only element of order $4$.
Then we consider the map $\varphi'_1(1_4) \to \sigma_2$.
Then $\mathbb{Z_5} \rtimes_{\varphi'_1} \mathbb{Z_4}$