A two variable Rankin-Selberg integral for $\mathrm{G}\mathrm{U}(2,2)$ and the degree 5 $L$-function of ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$

We give a two-variable Rankin–Selberg integral for generic cusp forms on ${\mathrm{P}\mathrm{G}\mathrm{L}}_4$ and ${\mathrm{P}\mathrm{G}\mathrm{U}}_{2,2}$ which represents a product of exterior square $L$-functions. As a residue of our integral, we obtain an integral representation on ${\mathrm{P}\mathrm{G}\mathrm{U}}_{2,2}$ of the degree 5 $L$-function of ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ twisted by the quadratic character of $E\mathrm{/}F$ of cuspidal automorphic representations which contribute to the theta correspondence for the pair $({\mathrm{P}\mathrm{G}\mathrm{S}\mathrm{p}}_4,{\mathrm{P}\mathrm{G}\mathrm{U}}_{2,2})$.