Let $M$ be a $2$-torsion free $\sigma$-prime $\Gamma$-ring and $U$ be a non-zero $\sigma$-square closed Lie ideal of $M$. If $T :M\rightarrow$ $M$ is an automorphism on $U$ such that $T\ne 1$ and $T\sigma =\sigma T$ on $U$, then we prove that $U\subseteq Z(M)$. We also study the additive maps $d : M\rightarrow M$ such that $d(u\alpha u)=2u\alpha d(u)$, where $u\in U$ and $\alpha\in\Gamma$, and show that $d(u\alpha v)=u\alpha d(v) + v\alpha d(u)$, for all $u,v\in U$ and $\alpha\in\Gamma$.