Definition:G-Submodule

Definition
Let $\left({G, \cdot}\right)$ be a finite group.

Let $\left({V, \phi}\right)$ be a $G$-module.

Let $W$ be a vector subspace of $V$.

If $\phi$ is a linear group action when restricted to $G \times W \subseteq G \times V$.

Then $(W,\phi_W)$; where $\phi_W$ is the restriction of $\phi$ on $G\times W$, is called a $G$-submodule of $(V,\phi)$.

Remark
In G-Submodule Test it is proven that:

$W$ is a $G$-submodule if $\phi(G,W)\subseteq W$

This is a much easier to check than the raw definition.