In the interest of pursuing the analogies between finite groups and (finite-dimensional) Lie groups, it seems natural to call the Schur multiplier of a finite group analogous to the fundamental group of a Lie group. Just as the Schur multiplier limits what groups can arise as central subgroups of a group $G$ with fixed (isomorphism type of) $Inn(G)$, does the fundamental group do the same thing for Lie groups?
One baby example of this question: If $G$ is a Lie group with a central subgroup $Z$ with $|Z|=4$ and $G/Z \cong SO(3)$, does it follow that $G$ has a subgroup of index $2$?
Also, in the above situation, if $G$ has no subgroup isomorphic to $SO(3)$ and $Z$ is cyclic, does it follow that $G \cong H $, where $H$ is the subgroup of $U(2)$ obtained by adjoining $i$ times the identity matrix to $SU(2)$?
