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

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