I want to show if $f \in K_4=\{id,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)\}$, then for all permutations $h \in S_4$ also $h^{-1}fh \in K_4$ (not using group theory).
The elements in $K_4$ except for the identity are the elements in $S_4$ which are a product of two disjoint 2-cycles.
Therefore, I am trying to prove that $h^{-1}fh$ in disjoint cycle notation are two 2-cycles.
Then I consider different cycle structures of $h$, $4$ 1-cycles, one 3-cycle and one 1-cycle etc.
However, I am not sure why it is true for instance if $h$ is $(1)(2 3 4)$, taking the $f$ to be the identity I get $(1)(2)(3 4)$ which is not two 2-cycles.