  Let $\Omega$ be a group with identity $e$, $\Gamma$ be a $\Omega$-graded
commutative ring and $\Im$ a graded $\Gamma$-module. In this article, we
introduce the concept of $gr$-$C$-$2^{A}$-secondary submodules and investigate
some properties of this new class of graded submodules. A non-zero graded
submodule $S$ of $\Im$ is said to be a $gr$-$C$-$2^{A}$-secondary submodule if
whenever $r,s \in h(\Gamma)$, $L$ is a graded submodule of $\Im$, and
$rs\,S\subseteq L$, then either $r\,S\subseteq L$ or $s\,S \subseteq L$ or $rs
\in Gr(Ann_\Gamma(S))$.
