  Let $p$ be an odd prime and $e_p$ be its irregularity index. If $4e_p+8
<\frac{p-1}{2}$ we construct a Galois representation with image in the diagonal
torus of $\op{GSp}_4(\Fp)$ that lifts to a characteristic $0$ representation
unramified at all primes $\ell\neq p$ with image containing a finite index
subgroup of $\Symp$.
