I'm in the process of trying to find $|Aut(K_{4})|$, where $K_{4}$ is the Klein-4 group, isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$. $K_{4} = \{ id, (12)(34), (13)(24), (14)(23)\}$ is a subgroup of $S_{4}$, the group of permutations of $4$ objects.
Right now, I have already established that $K_{4} \simeq \mathbb{Z}_{2} \times \mathbb{Z}_{2}$ and have proven that if $G,H$ groups such that $G \simeq H$, then $Aut(G) \simeq Aut(H)$.
So, I am at the point where I am ready to check which of the permuations in $K_{4}$ are homorphisms, and I am unsure of how to do that.
For example, for the permuation $(12)(34) = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{pmatrix}$, I want to show that the map $f: \begin{pmatrix}1 & 2 & 3 & 4 \end{pmatrix} \mapsto \begin{pmatrix} 2 & 1 & 4 & 3 \end{pmatrix} $ is a homomorphism, but I am not sure what the appropriate group operations to use are.
Until now, I have only ever calculated $Aut$ of $\mathbb{Z}_{n}$, where the operation I would use was addition modulo $m$, but I've never dealt with a case where the underlying group was a permutation group and not an additive group. Could somebody please demonstrate for me how to show that $f: \begin{pmatrix}1 & 2 & 3 & 4\end{pmatrix} \mapsto \begin{pmatrix} 2 & 1 & 4 & 3 \end{pmatrix} $ meets the criteria for being homomorphic? If I see one worked out example, I can probably figure out how to do the rest on my own, but as it is right now, I am at a loss.
Thank you.