  Generalizing classical extension theory, we solve a Schreier-type extension
problem for polygroups by groups. As a consequence, we obtain a method for
computing a presentation for a group from its action on a set. The usefulness
of this method is illustrated by deriving explicit presentations for the groups
$GL_2$ over valuation rings and over valued fields, for the groups $SL_3$ over
arbitrary fields, as well as for the five Mathieu groups. Moreover, we sketch
some aspects of a new deformation technique for groups, their actions, and
presentations, and apply it to compute presentations for the sharply
$3$-transitive Zassenhaus groups $M(q^2)$ (in the notation of Huppert and
Blackburn) for any odd prime power $q$. This computation serves to demonstrate
how suitable deformation of groups and their actions interacts with, and
thereby enhances, the presentation method.
