• problem with torsion in $\pi_1(G)$.

So, what's the difference between $GL_2, PGL_2, SL_2$? $GL_2$ is the father of all, $PGL_2$ and $SL_2$ each take out some abelian part out of it. $PGL_2$ totally killed the central subgroup, by quotienting out it, whereas $SL_2$ tries to take a slice, that intersects minimally with it.