In this thesis a particular class of monoid extensions are studied and characterized, the weakly Schreier split extensions. It is demonstrated that both Artin glueings of frames and λ-semidirect products of inverse semigroups are examples of these extensions. The characterization is given in terms of a generalization of an action. This makes the theory amenable to cohomological ideas. Specifically, a new class of extensions called cosetal extensions are introduced and characterized. When parameterized by this new notion of action, a Baer sim may be defined in these extensions giving rise to an analogue of the second cohomology group. Finally, a connection is made to the setting of toposes, exploiting the link between toposes and frames.