Epimorphisms of demushkin groups

A necessary condition that a continuous epimorphism from a Demushkin groupG onto a finitep-groupH can be factored epimorphically through a free prop-groupS-is given, which is sufficient whenH is abelian of exponentp ^m≠2,m depending onG, 1≦m≦∞. In particular a free prop-factor groupS toG can have rank at most one half rankG. Application is made to embedding problems over localp-adic fields.