$p$-adic $L$-functions for $\mathrm{G}\mathrm{S}\mathrm{p}(4)\times \mathrm{G}\mathrm{L}(2)$ II

We construct four-variable $p$-adic $L$-functions for cuspidal Hida families on $\mathrm{G}\mathrm{S}\mathrm{p}(4)\times \mathrm{G}\mathrm{L}(2)$ and prove a complete interpolation formula. The archimedean zeta integrals are computed by using a partial interpolation formula for the four-variable $p$-adic $L$-functions and some previously constructed $p$-adic $L$-functions.