Homomorphism from $S_4$ to $S_6$ I was asked the question

There are $6$ subsets of order $2$ of $\{1,2,3,4\}$. Any element of $S_4$ permutes these subsets. Show that the resulting homomorphism from $S_4$ to $S_6$ is injective and its image lies in $A_6$.

Now, I can understand that any element of $S_4$ permutes some of the elements of order $2$, and that gives us a permutation of $S_6$. But, I can't seem to have a strong grasp of how the permutation of $S_4$ is influencing the permutation of $S_6$. So, I can't figure out why two different permutations of $S_4$ can't take us to the same permutation of $S_6$. Please help in this.
Also, I can see that since we are only considering the two element subsets, there's already a sense of parity introduced in the system. But, how can we mathematically show that all of these permutations of $S_4$ only gives us even permutations?
 A: Let's prove injectivity first, since it is the easier claim.
Given some non-identity permutation $\sigma\in S_4$, consider $\sigma(\{1,2\}),\;\sigma(\{3,4\})$. If they aren't equal to $\{1,2\},\{3,4\}$ respectively, then $\sigma$ is mapped to a non-identity permutation as we desired. So assume that they are equal.
Then, either $\sigma(1)=2,\;\sigma(2)=1$ or $\sigma(3)=4,\;\sigma(4)=3$. Assume the former without loss of generality.
Then, $\sigma(3)\neq1$, so $1\notin\sigma(\{1,3\})$, which proves that $\sigma$ is not mapped to the identity permutation. This shows that the mapping is injective.
For the other part, show that the transposition that sends $1\to2$ and $2\to1$ maps to an even permutation. Thus, every transposition maps to an even permutation. Then, prove that the mapping respects composition, and you are done.
Proof of the mapping respecting composition:
Let $\sigma,\gamma\in S_4$, and let $F$ be the mapping in question.
For any $i\neq j\in\{1,2,3,4\}$, $$F(\sigma\circ\gamma)(\{i,j\})=\{\sigma(\gamma(i)),\sigma(\gamma(j))\}$$$$F(\sigma)\circ F(\gamma)(\{i,j\})=F(\sigma)(\{\gamma(i),\gamma(j)\})=\{\sigma(\gamma(i)),\sigma(\gamma(j))\}$$
A: HINT:

*

*Every subset with $1$ element is the intersection of two subsets with $2$ elements.


*Show that every transposition maps to an even permutation.
