= Solution
The <sign homomorphism> $\operatorname{sgn}:S_n\to\{+1,-1\}$ records <parity of a permutation>. The <kernel of a group homomorphism> in this case is the <alternating group> $A_n$, consisting of even <permutations>. For $n\geq2$ it has index two in the <symmetric group> and order $n!/2$.
The <cycle type> is the multiset of lengths of the disjoint cycles, including fixed points if desired. For $A_4$ the complete count is
$$
\begin{array}{c|c|c}
\text{cycle type}&\text{kind}&\text{number}\\\hline
1^4&\text{identity}&1\\
(3,1)&\text{three-cycle}&\binom43\,2=8\\
(2,2)&\text{two disjoint transpositions}&3
\end{array}
$$
A single <transposition> and a four-cycle are odd and are excluded. Thus
$$
\boxed{|A_4|=12,\quad\text{with cycle types }1^4,\ (3,1),\ (2,2).}
$$
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