In this proof:enter link description here
Page 19, it gives a construction of outer automorphism of $S_6$,it sends $S_6$ to Perm($S_6$/H)= $S_6$, and by the former proof, it is injective.However, it sends subgroup H to (1).Thus in this sense, it seems not injective. I feel so confused.
There are $6$ cosets $Hg_1,\cdots,Hg_6$ of $H$ in $G$, and multiplying each of these on the right by some $g \in G$ will permute these cosets. Thus we have the homomorphism $\phi:G\to \text{Sym}(\{gH\})$ sending $g$ to its permutation on the cosets. Note that although $e$ does not permute the cosets, all other elements of $h \in H$ will do permute the cosets nontrivially, because $Hg_ih$ is not necessarily equal to $Hg_i$. The proof of why the $h$ do not permute the cosets is, I think, implicit in the text.