Any suggestions/methods/estimates for the following problem would be very appreciated.
$l,p$ are primes with $p \equiv 1 \!\! \pmod l$. $r$ is a positive integer with $r \equiv 1 \!\! \pmod p$ and $q = p^r$. $Tr$ is the field trace $\mathbb{F}_q \to \mathbb{F}_p$, so $Tr(x) = x + x^p + \cdots + x^{p^{r-1}}$. Note that $Tr\,|_{\mathbb{F}_p} = \text{id}$.
Finally $H = H_1$ is the subgroup of order $l$ elements of $\mathbb{F}_p^{\times}$ and $H_r$ is the largest subgroup of $\mathbb{F}_q^{\times}$ such that $H \subset H_r$ and $Tr(H_r) \subset H$.
Questions around $H_r$: How big is $H_r$? Or, varying $r$, how big can $H_r$ be made relative to $\mathbb{F}_q^{\times}$? Can anything be said about the size of the fibers of $Tr: H_r \to H$? Can they be made to have the same size?
(The idea behind this is that I'd like to be able to compare an exponential sum over $H$ with one over $H_r$.)
(Note that $H_r = H + K_r$ for some subset $K_r$ of $\ker Tr = \{ x \in \mathbb{F}_q \: | \: Tr(x) = 0$}.)