TITLE: Extensions of $SL(2,\mathbb{F}_q)$
QUESTION [6 upvotes]: Let $n = q(q^2-1)$ (the order of $SL(2,\mathbb{F}_q)$ I think).  How many groups of order $2n$ contain $SL(2,\mathbb{F}_q)$ as a (necessarily normal) subgroup?  Is this number known exactly?  Seems like it would be something well known, but maybe I slept through Group Theory class that day?

REPLY [8 votes]: I can give a complete answer to the question, but it is indeed a little delicate. For the moment I am not going to give much explanation, and there are places where I haven't got a completely formal proof of what happens.
We first distinguish the possible extensions by the outer automorphism of ${\rm SL}_2(q)$ induced by the external element. Let $q = p^e$ with $p$ prime. There are four possibilities for this outer automorphism:

Trivial.
Diagonal automorphism ($q$ odd only).
Field automorphism ($e$ even only).
Field $\times$ diagonal ($q$ odd and $e$ even).

In fact Case 4 does not occur. The first instance of this is $q=9$, where it is "well known" that there is no group with structure $2 ^. A_6 .2_3$ in the ATLAS notation.
For Case 1, there is a single isomorphism type of extensions (i.e. $C_2 \times {\rm SL}_2(q)$) when $q$ is even, and two isomorphism types (i.e. the direct product, and the central product of ${\rm SL}_2(q)$ with $C_4$), when $q$ is odd.
In Case 2, with $q$ odd, there are two isomorphism types of extensions. They both have $G/Z(G) \cong {\rm PGL}_2(q)$, and they can be distinguished by their Sylow $2$-subgroups. The first type has two conjugacy classes of elements of order $2$ and semidihedral Sylow $2$-subgroup, whereas the second type has a unique class of elements of order $2$ (the central element of the group) and generalized quaternion Sylow $2$-subgroups.
So this answers your question above for the case $q=p$ with $p$ odd. There are actually four isomorphism types of extensions in that case, two from case 1 and two from Case 2.
For Case 3 (field automorphisms of order $2$, with $e$ even), there is clearly a single isomorphism type of extensions, the semidirect product, when $q$ is even.
When $q$ is odd and $e$ is even, there appears to be again a unique isomorphism type, but this is less obvious. As in Case 2, there are two equivalence classes of extensions of $C_2$ by ${\rm PSL}_2(q).2$ in this case, but this time they are isomorphic as groups, and the two equivalence classes are interchanged by the diagonal automorphism of ${\rm PSL}_2(q)$.