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?
This community wiki solution is intended to clear the question from the unanswered queue.
As Moishe Cohen pointed out in his comment, the phrase "A frequently occurring case is a double covering group, a topological double cover in which $H$ has index $2$ in $G$" appeared for the first time in the January 2009 revision of the Wikipedia article and it is wrong.
In general $H$ is not a subgroup of $G$, and it does not even embed as a subgroup into $G$. Therefore it does not make sense to speak about the index of $H$ in $G$.
Only in the trivial case $G = H \times \mathbb{Z}_2$ there would be a reasonable interpretation.