By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: http://www.jmilne.org/math/xnotes/tc.pdf) we know that a given affine group scheme G can be recovered from their category of representations Rep(G) (thought of as a neutral Tannakian category). If G-->G' is a morphism Prop. 2.21 characterizes when is it faithfully flat (resp. closed immersion) in terms of properties of the induced map Rep(G')-->Rep(G).

My question is the following: is it possible to characterize that the sequence K-->G-->G' is exact in terms of the induced maps Rep(G')-->Rep(G)-->Rep(K).

( Actually I am interested in a situation when Rep(G)-->Rep(G') is known and one wonders if one can "construct" Rep(K) ).

By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: http://www.jmilne.org/math/xnotes/tc.pdf) we know that a given affine group scheme G can be recovered from their category of representations Rep(G) (thought of as a neutral Tannakian category). If G-->G' is a morphism Prop. 2.21 characterizes when is it faithfully flat (resp. closed immersion) in terms of properties of the induced map Rep(G')-->Rep(G).

My question is the following: is it possible to characterize that the sequence K-->G-->G' is exact in terms of the induced maps Rep(G')-->Rep(G)-->Rep(K).

( Actually I am interested in a situation when Rep(G')-->Rep(G) is known and one wonders if one can "construct" Rep(K) ).