Permutation groups containing infinite linear groups and reducts of infinite dimensional linear spaces over the two element field

Let ${\mathbb{F}}_2^{\omega }$ denote the countably infinite dimensional vector space over the two element field and $\mathrm{GL}(\omega ,2)$ its automorphism group. Moreover, let $\mathrm{Sym}({\mathbb{F}}_2^{\omega })$ denote the symmetric group acting on the elements of ${\mathbb{F}}_2^{\omega }$. It is shown that there are exactly four closed subgroups, $G$, such that $\operatorname{GL}(\omega, 2)\leq G\leq \operatorname{Sym}($\mathbb{F}$_2^\omega)$\mathrm{.}As$$\mathbb{F}$_2^\omega$isan$\omega$-categorical (and homogeneous) structure, these groups correspond to the first order definable reducts of ${\mathbb{F}}_2^{\omega }$. These reducts are also analyzed. In the last section the closed groups containing the infinite symplectic group $\mathrm{Sp}(\omega ,2)$ are classified.