Counting morphisms between specified groups

#group_theory

Determine the number of group homomorphisms $ϕ$ between the given groups. Here $K_{4}$ denotes the Klein four-group (also known as $Z / 2 Z \times Z / 2 Z$ ) and $S_{3}$ denotes the symmetric group on three elements.

$ϕ : K_{4} \to Z / 2 Z$
$ϕ : Z / 2 Z \to K_{4}$
$ϕ : S_{3} \to K_{4}$
$ϕ : K_{4} \to S_{3}$

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Determine the number of group homomorphisms $\phi$ between the given groups. Here $K_4$ denotes the Klein four-group (also known as ${\bf Z}/2{\bf Z}\times {\bf Z}/2{\bf Z}$) and $S_3$ denotes the symmetric group on three elements.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item $\phi:K_4\to {\bf Z}/2{\bf Z}$
	\item $\phi:{\bf Z}/2{\bf Z}\to K_4$
	\item $\phi:S_3\to K_4$
	\item $\phi:K_4\to S_3$
\end{enumerate}