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Let $H$ be a group, $m$ be a positive integer, $Ext_m H$ be the set of allisomorphic in $G$ classes of group monomorphisms $\varphi: H \rightarrow G$such that index of $\varphi(H)$ in $G$ is $m$. The main goal of this paper isto describe the elements of $Ext_m H$ in terms of a new concept of hypergroupsover the group. The obtained result is a very broad generalization of theSchreier theorem (1926). As an application, a series of intermediategeneralizations is obtained; particularly, a description of the set $Ext (H,Q)$ of isomorphic classes of all extensions of a noncommutative group $H$ by$Q$.

Hypergroups over the group and generalizations of Schreier's theorem on group extensions