I am not very familiar with group extensions, and I know that in general finding all possible extensions is an extremely hard problem, however I am interested in the following:
What are all possible extensions of a finite Möbius group $H$ by the cyclic group on two elements $C_2$. ie all groups $G$ such that there are homomorphisms such that
$$ 1 \longrightarrow C_2 \longrightarrow G \longrightarrow H \longrightarrow 1 $$
is a short exact sequence (since there is some confusion about what I mean).
This is of interest to me because it gives all possible automorphism groups of hyperelliptic surfaces, and I would like to be able to explain how we determine what these groups are. I have a list of the possible groups but I would like to know if they are equal to the set of extensions, or a proper subset of the possible extensions.
Edit: To be clear here, I am interested in the group theory behind finding these extensions, not in the automorphisms of certain curves, I already understand those.
A reference for this problem would also be a helpful answer. The only resource I have used so far is Milne's notes on group theory and they do not discuss how to solve this sort of problem.
EDIT It may be useful here to know that we can prove using properties of Riemann surfaces that $C_2$ is in the centre of the groups of interest. So I know the groups of interest will be central extensions. According to wikipedia the set of isomorphism classes of such groups are exactly the cohomolgy group $H^2(G,A)$ which is useful. I guess this answers my question if I restrict it accordingly.