Extensions of algebraic groups with finite quotient and nonabelian 2-cohomology

For a finite smooth algebraic group F over a field k and a smooth algebraic group G¯ over the separable closure of k, we define the notion of F-kernel in G¯ and we associate to it a set of nonabelian 2-cohomology. We use this to study extensions of F by an arbitrary smooth k-group G. We show in particular that any such extension comes from an extension of finite k-groups when k is perfect and we give explicit bounds on the order of these finite groups when G is linear. We prove moreover some finiteness results on these sets.