  It is well known that the set of isomorphism classes of extensions of groups
with abelian kernel is characterized by the second cohomology group. In this
paper we generalise this characterization of extensions to a natural class of
extensions of monoids, the cosetal extensions. An extension k: N -> G, e: G ->
H, where k is the inclusion and e is the quotient , is cosetal if for all g,g'
in G in which e(g) = e(g'), there exists a (not necessarily unique) n in N such
that g = k(n)g'. These extensions generalise the notion of special Schreier
extensions, which are themselves examples of Schreier extensions. Just as in
the group case where a semidirect product could be associated to each extension
with abelian kernel, we show that to each cosetal extension (with abelian
group) kernel, we can uniquely associate a weakly Schreier split extension. The
characterization of weakly Schreier split extensions is combined with a
suitable notion of a factor set to provide a cohomology group granting a full
characterization of cosetal extensions, as well as supplying a Baer sum.
