We estimate double sums
$$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$ with a multiplicative character
${\it\chi}$ modulo
$p$ where
${\mathcal{I}}=\{1,\dots ,H\}$ and
${\mathcal{G}}$ is a subgroup of order
$T$ of the multiplicative group of the finite field of
$p$ elements. A nontrivial upper bound on
$S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$ can be derived from the Burgess bound if
$H\geq p^{1/4+{\it\varepsilon}}$ and from some standard elementary arguments if
$T\geq p^{1/2+{\it\varepsilon}}$, where
${\it\varepsilon}>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters
$H$ and
$T$. We also estimate double sums
$$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$ and give an application to primitive roots modulo
$p$ with three nonzero binary digits.