I was asked to show that $\text{Aut}(K_n)$ is isomorphic to $\text{Aut}(L(K_n))$ if and only if $n\neq2,4$.
Here, $K_n$ is the complete graph on $n$ vertices, and $L(K_n)$ is the line graph of the complete graph. Also, $\text{Aut}(L(K_n))$ is called the edge automorphism group of $K_n$.
For the $\impliedby$ case. a simple enumeration shows that the automorphism groups of $K_2$ and $K_4$ are not isomorphic to their edge automorphism groups.
Homomorphism between the two groups is easy to verify. Now, suppose $g$ is an automorphism of $K_n$, it induces an automorphism (since incidency of edges is preserved) on $L(K_n)$.
I am not certain how to proceed from here or how this exactly fails for the case $n=2,4$ and would greatly appreciate any help.