  We give a geometric construction of the Heisenberg-Weil representation of a
finite unitary group by the middle \'{e}tale cohomology of an algebraic variety
over a finite field, whose rational points give a unitary Heisenberg group.
Using also a Frobenius action, we give a geometric realization of the Howe
correspondence for $(\mathit{Sp}_{2n},O_2^-)$ over any finite field including
characteristic two. As an application, we show that unipotency is preserved
under the Howe correspondence.
