On some linear groups, having a big family of $G$-invariant subspaces

Let $F$ be a field, $A$ a vector space over $F$, $GL(F, A)$ be the group of all automorphisms of the vector space $A$. If $B$ is a subspace of $A$, then denote by $BFG$ the $G$-invariant subspace, generated by $B$. A subspace $B$ is called nearly $G$-invariant, if $dim_F(BFG/B)$ is finite. In this paper we described the situation when every subspace of $A$ is nearly $G$-invariant.