It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any reduced scheme over a perfect field.

I have a counterexample for non-reduced schemes, but how about the perfectness of the base field? So my questions are:

(1) Where does Serre's proof brakes down for non-perfect base fields?

(2) Does anyone know a counterexample over non-perfect fields?

NB: Serre does not state the hypothesis (maybe because at that time varieties were always integral over an algebraically closed field), but every author who quotes him states those hypothesis.