Fixed points of the action of an algebraic group

Linearly reductive groups, that is, linear algebraic groups, say over a field, in which the functor of invariants from finite dimensional representations to vector spaces is exact. I am not sure this is in the literature in this generality, but it is not so hard to prove with a formal scheme argument.

Also, I would conjecture that this is optimal, that is, given a non linearly reductive algebraic group, one can find a smooth variety on which this acts, such that the fixed point locus is not smooth.

Angelo's conjectural statement is proven in Fogarty, J.; Norman, P. "A fixed-point characterization of linearly reductive groups." MR0485896

(not enough reputation to post this as comment, sorry)