  This paper is a complement to the work of the second author on modular
quotient singularities in odd characteristic (see arXiv:1210.8006). Here we
prove that if $V$ is a three-dimensional vector space over a field of
characteristic $2$ and $G<GL(V)$ is a finite subgroup generated by
pseudoreflections and possessing a $2$-dimensional invariant subspace $W$ such
that the restriction of $G$ to $W$ is isomorphic to the group
$SL_{2}(\mathbb{F}_{2^n})$, then the quotient $V/G$ is non-singular. This,
together with earlier known results on modular quotient singularities, implies
first that a theorem of Kemper and Malle on irreducible groups generated by
pseudoreflections generalizes to reducible groups in dimension three, and,
second, that the classification of three-dimensional isolated singularities
which are quotients of a vector space by a linear finite group reduces to
Vincent's classification of non-modular isolated quotient singularities.
