  We classify the orbits of solids in the projective space $\text{PG}(5,q)$,
$q$ even, under the setwise stabiliser $K \cong \text{PGL}(3,q)$ of the
Veronese surface. For each orbit, we provide an explicit representative $S$ and
determine two combinatorial invariants: the point-orbit distribution and the
hyperplane-orbit distribution. These invariants characterise the orbits except
in two specific cases (in which the orbits are distinguished by their
line-orbit distributions). In addition, we determine the stabiliser of $S$ in
$K$, thereby obtaining the size of each orbit. As a consequence, we obtain a
proof of the classification of pencils of conics in $\text{PG}(2,q)$, $q$ even,
which to the best of our knowledge has been heretofore missing in the
literature.
