Cameron-Liebler line classes are sets of lines in PG(3,q) that contain a
fixed number $x$ of lines of every spread. Cameron and Liebler classified them
for $x\in\{0,1,2,q^2-1,q^2,q^2+1\}$ and conjectured that no others exist. This
conjecture was disproved by Drudge and his counterexample was generalised to a
counterexample for any odd $q$ by Bruen and Drudge.
In this paper, we give the first counterexample for even $q$, a Cameron-Liebler
line class with parameter $7$ in PG(3,4). We also prove the nonexistence of
Cameron-Liebler line classes with parameters $4$ and $5$ in PG(3,4) and give
some properties of a hypothetical Cameron-Liebler line class with parameter $6$
in PG(3,4).