Image of mod p Galois Representation of Jacobian of genus 2 curve (awaiting review)

WARNING: The image types of nonmaximal primes is still in development, and subject to change.

Let $p$ be a prime and let $C$ be a genus 2 curve defined over $\mathbb{Q}$.

Subgroups $G$ of $\GSp(4,\F_\ell)$ that can arise as the image of the mod-$\ell$ Galois representation \[ \rho_{J,p}\colon {\Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \GSp(4,\F_\ell) \] attached to the jacobian $J$ of $C$ that do not contain $\Sp(4,\F_\ell)$ are identified via the types arising from Mitchell's 1914 classification.

There are six types: Irreducible irred, Cuspidal cusp, One-plus-Three 1p3, Two-plus-Two 2p2, Non-Semistable nss, and last but very much not least, the super duper mysterious and esoteric unknown type ?. Can you catch 'em all?