A two variable Rankin-Selberg integral for GU(2,2) and the degree 5 L-function of GSp₄

We give a two-variable Rankin--Selberg integral for cusp forms on PGL₄ and PGU2,2 which represents a product of exterior square L-functions. As a residue of our integral, we obtain an integral representation on PGU2,2 of the degree 5 L-function of GSp₄ twisted by the quadratic character of $E/F$ of cuspidal automorphic representations which contribute to the theta correspondence for the pair (PGSp₄,PGU2,2).