For a class of affine algebraic groups C over a field, we define the notions of C-fundamental gerbe of a fibered category, generalizing what we had done in arXiv:1204.1260 for finite group schemes. We give sufficient conditions on C implying that a fibered category X over κ satisfying mild hypotheses admits a Nori C
Nearby in the stack
-fundamental gerbe. We show that these are verified in particular by the classes of virtually abelian and virtually unipotent group schemes. In the second situation, under a properness condition on
X
, we give a tannakian interpretation of the resulting gerbe.