18.60 (2014)

Let $V$ be an infinite countable elementary abelian additive 2-group. Does $\text{Aut}\,V$ contain a subgroup $G$ such that
$\qquad$ a) $G$ is transitive on the set of non-zero elements of $V$, and
$\qquad$ b) if $H$ is the stabilizer in $G$ of a non-zero element $v \in V$, then $V = \langle v \rangle \oplus V_v$, where $V_v$ is $H$-invariant, $H$ is isomorphic to the multiplicative group $P^*$ of a locally finite field $P$ of characteristic 2, and the action $H$ on $V_v$ is similar to the action of $P^*$ on $P$ by multiplication?

Conjecture: such a group $G$ does not exist. If so, then the group $G$ in 18.59 does not exist too.