Please, please I ask that you do not refer me to the question that has been answered already: Sylow $2$-subgroups of $S_4.$ I do not understand the bisections and what they are describing in that explaination.
Rather, I need help with a different method to finding the other 2 subgroups by way of using $\sigma H \sigma^{-1},$ and using the fact that $\sigma H \sigma^{-1},= (\sigma(a1); \sigma(a2); . . . ; \sigma(ak)).$
I am just having a hard time picking an element in $S_4$ and using this formula to produce something that doesnʻt look like $H$.
They do "look like" $H$, because they are isomorphic to $H$.
To find them, just play around with some elements $g$ of $S_4$, computing, say, $g(12)g^{-1}$, until you get something not in $H$.
Once you find such a $g$, one of the other Sylow subgroups will be $gHg^{-1}$.
Of course, if $g$ consists of cycles disjoint from $(12)$, you'll just get $(12)$ back again. So choose something else, until you find one that works.
This works because conjugation is an automorphism.