The privileges of a Ground Instructor are spelled out in part 61.215
§ 61.215 Ground instructor privileges.

top
(a) A person who holds a basic ground instructor rating is authorized to
blah blah blah does not apply to the question.
(b) A person who holds an advanced ground instructor rating is authorized to provide:
(1) Ground training on the aeronautical knowledge areas required for the issuance of any certificate or rating under this part except for the aeronautical knowledge areas required for an instrument rating.
So he can give ground school for a written except the Inst.
(2) The ground training required for any flight review except for the training required for an instrument rating.
He can give the 2 hours of ground training for a flight review.
(3) A recommendation for a knowledge test required for the issuance of any certificate or rating under this part except for an instrument rating.
He can sign you off for any part 61 written except the Inst.
(c) A person who holds an instrument ground instructor rating is authorized to provide:
(1) Ground training in the aeronautical knowledge areas required for the issuance of an instrument rating under this part;
He can give the ground school for the Inst written.
(2) Ground training required for an instrument proficiency check;
Well great except when you look in 61.57 under IPC there's no requirement for any ground training.
(3) A recommendation for a knowledge test required for the issuance of an instrument rating under this part.
And he can sign you off for Inst written.

So that's what an AGI/IGI can do. They teach and sign off ground school.