Let $G$ be a multiplicative group with identity $e$, $R$ be a $G$-graded commutative ring and $M$ be a graded $R$-module. The aim of this article is some investigations of graded $2$-absorbing submodules over $Gr$-multiplication modules. A graded submodule $N$ of $R$-module $M$ is called graded $2$-absorbing if whenever $a,b\in h(R)$ and $m\in h(M)$ with $abm\in N$, then either $ab\in (N :_R M)$ or $am\in N$ or $bm\in N$. We also introduce the concept of graded classical $2$-absorbing submodule as a generalization of graded classical prime submodules and show a number of results in this class.