= Solution
Extend $\sigma\in S_n$ to a <permutation> $\widehat\sigma$ of $n+2$ letters that fixes the last two letters. Let $t=(n+1\ n+2)$, and let $\epsilon(\sigma)$ be $0$ or $1$ according as $\sigma$ is even or odd. Define
$$
\iota(\sigma)=\widehat\sigma\,t^{\epsilon(\sigma)}.
$$
The two factors have disjoint supports and commute. Their signs cancel, so $\iota(\sigma)\in A_{n+2}$. Additivity of <parity of a permutation> modulo two gives
$$
\iota(\sigma\tau)=\widehat\sigma\widehat\tau\,t^{\epsilon(\sigma)+\epsilon(\tau)}
=\iota(\sigma)\iota(\tau).
$$
Thus $\iota$ is a <group homomorphism>. Its restriction to the first $n$ letters recovers $\sigma$, making it injective. Its image is therefore a <subgroup> isomorphic to $S_n$:
$$
\boxed{S_n\cong\iota(S_n)\leq A_{n+2}.}
$$
This <group embedding> works for every positive $n$, including the trivial case $n=1$.
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