Let $S_6$ be equipped with the natural action on $\{1,2,\dots,6\}$. For $1\leq i,j\leq 6$, define subsets: $$X_{j\to i}:=\{\sigma\in G\,\mid \sigma(j)=i\}.$$ Let $X:=\{X_{j\to i}\mid 1\leq i,j\leq 6\}$. Suppose that $\psi\in \mathrm{Aut}(S_6)$. Consider the set: $$\psi X_{j\to i}:=\{\psi(\sigma)\mid\sigma\in X_{j\to i}\}.$$ If $\psi$ is inner, then the $\psi$ is a permutation on $X$.
Proof: Let $\psi(\sigma)=t\sigma t^{-1}$ for $t\in S_6$. Let $\sigma\in X_{j\to i}$. Consider $$\begin{aligned} \psi(\sigma)(t(j))&=t\sigma t^{-1}t(j)=t\sigma(j)=t(i), \end{aligned}$$ that is $\psi(\sigma)\in X_{t(j)\to t(i)}$.
Question: What does $\psi$ do to the $X_{j\to i}$ when it is an outer automorphism of $S_6$? Is any nice description available?
I can see that we don't in general have stability. For example, both $(12),(23)\in X_{4\to 4}$, but there is an outer automorphism that sends $(12)$ to $(12)(34)(56)$ and $(23)$ to $(13)(25)(46)$, so it splits up $X_{4\to 4}$.
Note that the $\psi X_{j\to i}$ are subsets of size $5!$ in $S_6$.
Thanks to @user297024 in the comments, something indeed can be said. We use $X_{ji}:=X_{j\to i}$ and $P_{ji}=\psi(X_{ji})$.
Note for any $1\leq k\leq 6$, $X_{kk}$ is a stabiliser subgroup $S_5\subset S_6$. The image of $X_{kk}$ under an outer automorphism, $P_{kk}=\psi(X_{kk})$ is a transitive $S_5\subset S_6$.
An element $t\in X_{ji}$ gives rise to an element $\psi(t)\in P_{ji}$ and this (or maybe its inverse) conjugates $P_{ii}$ and $P_{jj}$, and similarly there are nice conjugations between $P_{ij}$ and $P_{kl}$ in the cases $i=k$, $j=l$ and $i\neq k$, $j\neq l$.