  In order to study the problems of extending an action along a quotient of the
acted object and along a quotient of the acting object, we investigate some
properties of the fibration of points. In fact, we obtain a characterization of
protomodular categories among quasi-pointed regular ones, and, in the
semi-abelian case, a characterization of strong protomodular categories.
Eventually, we return to the initial questions by stating the results in terms
of internal actions.
