2-Cohomologies of the groups $SL(n,q)$

Let $G=SL(n,q)$, where $q$ is odd, $V$ be a natural module over $G$, and $L=S^2(V)$ be its symmetric square. We construct a 2-cohomology group $H^2(G,L)$. The group is one-dimensional over $\mathbf F_q$ if $n=2$ and $q\neq3$, and also if $(n,q)=(4,3)$. In all other cases $H^2(G,L)=0$. Previously, such groups $H^2(G,L)$ were known for the cases where $n=2$ or $q=p$ is prime. We state that $H^2(G,L)$ are trivial for $n\ge3$ and $q=p^m$, $m\ge2$. In proofs, use is made of rather elementary (noncohomological) methods.