  Let $H$ be a group and $E$ a set such that $H \subseteq E$. We shall describe
and classify up to an isomorphism of groups that stabilizes $H$ the set of all
group structures that can be defined on $E$ such that $H$ is a subgroup of $E$.
A general product, which we call the unified product, is constructed such that
both the crossed product and the bicrossed product of two groups are special
cases of it. It is associated to $H$ and to a system $\bigl((S, 1_S,\ast),
\triangleleft, \, \triangleright, \, f \bigl)$ called a group extending
structure and we denote it by $H \ltimes S$. There exists a group structure on
$E$ containing $H$ as a subgroup if and only if there exists an isomorphism of
groups $(E, \cdot) \cong H \ltimes S$, for some group extending structure
$\bigl((S, 1_S,\ast), \triangleleft, \, \triangleright, \, f \bigl)$. All such
group structures on $E$ are classified up to an isomorphism of groups that
stabilizes $H$ by a cohomological type set ${\mathcal K}^{2}_{\ltimes} (H, (S,
1_S))$. A Schreier type theorem is proved and an explicit example is given: it
classifies up to an isomorphism that stabilizes $H$ all groups that contain $H$
as a subgroup of index 2.
