Field Trace Identities

Question

Let $\mathbb{F}_q$ be the finite field with $p^n$ elements and consider the trace map $$\mbox{Tr}: \mathbb{F}_q\to \mathbb{F}_p,$$ where $$\mbox{Tr}(\alpha)=\alpha+\alpha^p+\alpha^{p^2}+\cdots +\alpha^{p^{n-1}}.$$ If $\varphi \in \mbox{Gal}(\mathbb{F}_q/\mathbb{F}_p)$, then $\mbox{Tr}\big( \varphi(\alpha)\big)=\mbox{Tr}(\alpha).$ I was wondering if there was any identity for something of the form $\mbox{Tr}\big( \alpha\, \varphi(\beta)\big),$ where $\varphi \in \mbox{Gal}(\mathbb{F}_q/\mathbb{F}_p)$ and $\alpha,\beta \in \mathbb{F}_q$. This is isn't so bad if either $\alpha$ or $\beta$ lie in the base field. But what about the case $\alpha,\beta \in \mathbb{F}_q\backslash \mathbb{F}_p$?