Given your favorite version of the Heisenberg group one can prove the Stone-Von Neumann theorem. It is then not to hard to construct a family of representations on a central extension of $Sp\left(2n, F\right)$ by $\mathbb{C}^{\times}$, where for me $F$ is a local field.
In a past project I constructed such a representation in the case when $n=1$ and $F$ is finite, and I would like to deduce this special construction from the general construction outlined in my first paragraph. I believe this reduces to showing that when $F$ is finite $H^2 \left( SL_2 \left(F\right) , \mathbb{C}^{\times} \right)$ is trivial when $F$ has odd order not equal to 9.
That said, I've been bouncing back and forth between books and google trying to find a proof of the triviality of this Schur multiplier.Indeed I found one in Schur's 1904 paper
"Uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen."
However I was hoping that in the more than 100 years since Schur published this there has been a less German and more modern treatment of the triviality of $H^2 \left(SL_2 \left(F\right), \mathbb{C}^{\times} \right)$. So I'm wondering, is there such a treatment? Perhaps, as an alternative to a reference, someone could provide a sketch of the proof.
Edit 1: In the Atlas of Groups one can find an algorithm to calculate the Schur multiplier of a finite group, given generators and relations. However, I'd hope that there is a less computational proof that better capitalizes on the specific nature of the group $SL_2 \left(F\right)$.
Edit 2:Geoff Robinson and Jack Schmidt have pointed out that the Schur multiplier is nontrivial in the case that $F$ has order 4 or 9. Hopefully, my revised question is answerable.