  In this paper, we classify all continuous Galois representations
$\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to
\mathrm{GL}_2(\overline{\mathbf{Q}}_p)$ which are unramified outside
$\{p,\infty\}$ and locally induced at $p$, under the assumption that
$\overline{\rho}$ is exceptional, that is, has image of order prime to $p$. We
prove two results. If $f$ is a level one cuspidal eigenform and one of the
$p$-adic Galois representations $\rho_f$ associated to $f$ has exceptional
residual image, then $\rho_f$ is not locally induced and $a_p(f)\neq 0$. If
$\rho$ is locally induced at $p$ and with exceptional residual image, and
furthermore certain subfields of the fixed field of the kernel of
$\overline{\rho}$ are assumed to have class numbers prime to $p$, then $\rho$
has finite image up to a twist.
