  Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a
graded unitary $R$-module. In this article, we introduce the concepts of graded
$\phi$-$2$-absorbing and graded $\phi$-$2$-absorbing primary submodules as
generalizations of the concepts of graded $2$-absorbing and graded
$2$-absorbing primary submodules. Let $GS(M)$ be the set of all graded
$R$-submodules of $M$ and $\phi: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be
a function. A proper graded $R$-submodule $K$ of $M$ is said to be a graded
$\phi$-$2$-absorbing $R$-submodule of $M$ if whenever $x, y$ are homogeneous
elements of $R$ and $m$ is a homogeneous element of $M$ with $xym\in
K-\phi(K)$, then $xm\in K$ or $ym\in K$ or $xy\in (K :_{R} M)$, and $K$ is said
to be a graded $\phi$-$2$-absorbing primary $R$-submodule of $M$ if whenever
$x, y$ are homogeneous elements of $R$ and $m$ is a homogeneous element of $M$
with $xym\in K-\phi(K)$, then $xm$ or $ym$ is in the graded radical of $K$ or
$xy\in (K:_{R}M)$. We investigate several properties of these new types of
graded submodules.
