  In this paper we give Rankin-Selberg integrals for the quasisplit unitary
group on four variables, $\mathrm{GU}(2,2)$, and a closely-related quasisplit
form of $\mathrm{GSpin}_6$. First, we give a two-variable Rankin-Selberg
integral on $\mathrm{GU}(2,2)$. This integral applies to generic cusp forms,
and represents the product of the exterior square (degree six) $L$-function and
the standard (degree eight) $L$-function. Then we give a set of integral
representations for just the degree six $L$-function on the quasisplit
$\mathrm{GSpin}_6$. The $\mathrm{GSpin}_6$ integrals are reinterpretations of
an integral originally considered by Gritsenko for Hermitian modular forms. We
show that they unfold to a model that is not unique, and analyze the integrals
via the technique of Piatetski-Shapiro and Rallis.
