  The present work introduces new perspectives in order to extend finite group
actions from surfaces to 3-manifolds. We consider the Schur multiplier
associated to a finite group $G$ in terms of principal $G$-bordisms in
dimension two, called $G$-cobordisms. We are interested in the question of when
a free action of a finite group on a closed oriented surface extends to a
non-necessarily free action on a 3-manifold. We show the answer to this
question is affirmative for abelian, dihedral, symmetric and alternating
groups. As an application of our methods, we show that every non-necessarily
free action of abelian groups (under certain conditions) and dihedral groups on
a closed oriented surface extends to $3$-dimensional handlebody.
