This paper completes the classification of nets of conics containing at least one double line in $\mathrm{PG}(2,q)$ for $q$ even. This classification contributes to the classification of partially symmetric tensors in $\mathbb{F}_q^3 \otimes S^2 \mathbb{F}_q^3$, $q$ even. The proof is obtained using geometric and combinatorial properties of the Veronese surface in 5-dimensional projective space over the finite field of even order. In particular, the orbits of planes in $\mathrm{PG}(5,q)$ that intersect the nucleus plane of the Veronese surface in at least one point are classified. As a result, it is shown that there are exactly $18$ equivalence classes of nets in $\mathrm{PG}(2,q)$, $q$ even, containing at least one double line, $9$ of which have an empty base.