  For a class of affine algebraic groups $\mathcal C$ over a field, we define
the notions of $\mathcal C$-fundamental gerbe of a fibered category,
generalizing what we had done in arXiv:1204.1260 for finite group schemes. We
give sufficient conditions on $\mathcal C$ implying that a fibered category $X$
over $\kappa$ satisfying mild hypotheses admits a Nori $\mathcal C$-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.
