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We estimate double sums with a multiplicative character 
$$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$
${\it\chi}$ 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 and give an application to primitive roots modulo 
$$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$
$p$ with three nonzero binary digits.
