In wikipedia I read:
In mathematics, a covering group of a topological group $H$ is a covering space $G$ of $H$ such that $G$ is a topological group and the covering map $p : G \rightarrow H$ is a continuous group homomorphism. The map $p$ is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which $H$ has index 2 in $G$; examples include the Spin groups, Pin groups, and metaplectic groups.
In this definition $H$ is not necessary a subgroup of $G$ (so "$H$ has index 2 in $G$" doesn't make sense in the usual way). So my question is: what is "the index of $H$ in $G$" in this context?