  It is a well known result in the covering groups that a subgroup $G$ of the
fundamental group at the identity of a semi-locally simply connected
topological group determines a covering morphism of topological groups with
characteristic group $G$. In this paper we generalize this result to a large
class of algebraic objects called topological groups with operations, including
topological groups. We also give the cover of crossed modules within
topological groups with operations.
