About $\text{Sym}(G) \setminus \text{Aut}(G)$

Question

In this post it's used $$\phi_g \in \text{Sym}(H) \setminus \text{Aut}(H)$$

Does the $\setminus$ stand for "set difference"? If yes, what this difference really represents between groups, please? I'm used to factoring groups, but that symbol is new to me. More than this: is $\text{Sym}(G)$ bigger than $\text{Aut}(G)$ since the former is made by bijections, while the latter by isomorphims?

This could explain why we could subtract the second set from the first one.

Thanks in advance

Answer 1

Every automorphism of $H$ permutes the elements of $H$, so can be considered an element of $\mbox{Sym}(H)$, but not every permutation of elements of $H$ is (induced by) an automorphism.