Let's suppose that $\operatorname{Out}(S_{n\ne 6})=1$ is not a known result. Let's denote with $\mathcal{C}_{(i_1,\dots,i_k)}$ the conjugacy class of $S_7$ corresponding to the cycle type $(i_1,\dots,i_k)$ (whence $i_1+\dots+i_k=7$). It turns out that:
$$\#\mathcal{C}_{(1,1,1,2,2)}=\#\mathcal{C}_{(1,2,2,2)}(=105)\tag 1$$
So, in principle, there could be a non-inner automorphism of $S_7$, say $\varphi$, such that:
$$\varphi(\mathcal{C}_{(1,1,1,2,2)})=\mathcal{C}_{(1,2,2,2)}\tag 2$$
What does actually prevent the existence of such a $\varphi$?