Yes they are.
In fact the formation of ${\rm Aut}_S(\omega)$ commutes by definition with base change, so you can reduce to $S=k$, and $S\to k={\rm id}$.
Then the fact that the $k$-group morphism $G\to {\rm Aut}_k(\omega)$ is an isomorphism is the reconstruction theorem in Tannaka theory, see Proposition 2.8 in the paragraph
"Recovering an affine group scheme from its representations"
in
Tannakian Categories
P. Deligne and J. S. Milne
in the volume
Hodge cycles, motives, and Shimura varieties.
Lecture Notes in Mathematics, 900.
http://www.springerlink.com/content/978-3-540-11174-0/#section=74407&page=1&locus=0
or
http://www.jmilne.org/math/xnotes/tc.html
ps : non-neutral fibre functor sounds very strange. A tannakian category is said neutral precisely when there exists a fibre functor ... Also, my advice would be to read Deligne-Milne's article before Deligne's article.