Let $A=\{\beta \in \mathbb F_{q^{2k}}: \beta - \beta^q \in \mathbb F_{q^2} \}$ be a subset of $\mathbb F_{q^{2k}}$.
Then $A$ is a subspace of $\mathbb F_{q^{2k}}$ over $\mathbb F_{q^2}$.
So we can easily know that $|A| \ge q^2$. But I dont know the exact value of $|A|$ which means a cardinality of a set $A$. I tried to use a field norm and trace, but it can be helpful to know the case that the characteristic $p$ does not divide $k$.
How can I calculate the cardinality?