Mod-5 Galois images from abelian surfaces

For $J$ an abelian surface, the Galois representation ${\varrho }_{J,\mathrm{\ell }}:\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})\to \mathrm{A}\mathrm{u}\mathrm{t}(J[\mathrm{\ell }])\simeq {\mathrm{G}\mathrm{S}\mathrm{p}}_4({\mathbb{F}}_{\mathrm{\ell }})$ is typically surjective, with smaller images indicating extra arithmetic structure. It is already known how to probabilistically compute whether ${\rho }_{J,\mathrm{\ell }}$ is surjective, and recent work by Chidambaram computes $\mathrm{im}{\rho }_{J,\mathrm{\ell }}$ for $\mathrm{\ell }=2,3$. We probabilistically compute ${\rho }_{J,5}$ for the Jacobians of 95% of genus 2 curves in the L-functions and Modular Forms Database (LMFDB) for which ${\rho }_{J,5}$ is not yet known. For the remaining Jacobians, we determine the order of the image and give a short list of candidate images.