Double covers of PGL(2,R), PGL(2,F) ??

Let R be the real numbers, or let F be a non-Archimedean
local field (if you don't know what local field means, then
just consider the reals R)

Are there any natural double covers of PGL(2,R) or PGL(2,F)
that you can think of? What I mean by double cover is
a group G that fits inside an exact sequence
1 --> Z/2Z ---> G ---> PGL(2,R) ---> 1
where Z is the integers.

I am looking for maybe a matrix group, or a non-linear
group (i.e. algebraic group that is not an affine variety).
I am not looking for the trivial extension
PGL(2,R) x Z/2Z.

Thank you for your help.

Sincerely,

James
.