Are all isomorphic simply transitive subgroups of $S_n$ conjugate?

Yes. If a group $G$ of order $n$ acts on a set $X$ of size $n$ transitively, then the stabilizer must be trivial, so $X$ must be isomorphic to the left regular representation of $G$. In particular there is a unique isomorphism class of such action, so any two such actions on $X$ must be conjugate.

(The left and right regular representations are isomorphic, if by the right regular representation you mean $\rho(g) x = xg^{-1}$. The isomorphism sends $x$ to $x^{-1}$.)

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Group Theory

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