  Let $G = GL(V)$ for an N-dimensional vector space $V$ over an algebraically
closed field k, and $G^{\theta}$ the fixed point subgroup of $G$ under an
involution $\theta$ on $G$. In the case where $G^{\theta} = O(V)$, the
generalized Springer correspondence for the unipotent variety of the symmetric
space $G/G^{\theta}$ was studied by last two authors, under the assumption that
ch k is odd. The definition of $\theta$, and of the associated symmetric space
given there make sense even if ch k = 2. In this paper, we discuss the Springer
correspondence for those symmetric spaces of even characteristic. We show that
if N is even, the Springer correspondence is reduced to that of symplectic Lie
algebras in ch k = 2, which was determined by Xue. While if N is odd, we show
that a very similar phenomenon as in the case of exotic symmetric space of
level 3 appears.
