Infinite dimensional linear groups with a spacious family of $G$-invariant subspaces

Let $F$ be a field, $A$ be a vector space over $F$, $GL (F, A)$ be the group of all automorphisms of the vector space $A$. If $B leq A$ then denote by $mathop{rm Core}_G (B)$ the largest $G$-invariant subspace of $B$. A subspace $B$ is called almost $G$-invariant if $mathop{rm dim}_F (B/mathop{rm Core}_G (B))$ is finite. In this paper we described the {case where} every subspace of $A$ is almost $G$-invariant.