I am looking for all groups $H$ that fit in to an exact sequence:
$0 \to Z/2Z \to H \to A_4 \to 1$, where here $A_4 \subset S_4$ are the even permutations.
I know two - there is one split extension, the direct product, and there is also the extension which is described by the action $SL(2,3)$ on $P^1_{F_3}$.
Are these all of them?
I am asking this because I want to understand what the cover in $SU(2)$ of the group of tetrahedral symmetries in $SO(3)$ is. I know in principle one can compute this directly, but I would like some direct explanation for the question posed above. (There is a unique element of order 2 in $SU(2)$, so this can be used to eliminate the trivial extension.)
I am also interested in this question in the situation where the groups on the right are $S_4$ and $A_5$, for the same reason.