On Pseudofunctors Sending Groups to 2-Groups

For a category $${\textsf{B}}$$ with finite products, we first characterize pseudofunctors from to $$\mathbb {C}\textsf{at}$$ whose associated opfibration is Cartesian monoidal. Among those, then the ones which extend internal groups 2-groups. If additive, this case precisely when has groupoidal fibres.