Explicit doubling integrals for $\widetilde{\mathrm{Sp}_2}(\mathbb{Q}_2)$ using ``good test vectors''

In a previous paper (see http:/www.math.ohio-state.edu/~czorn/works.html), we computed examples of the doubling integral for constituents of the unramified principal series of $\mathrm{Sp}_2(F)$ and $\widetilde{{\rm Sp}_2}(F)$ where $F$ was a non-dyadic field. These computations relied on certain ``good test vectors'' and ``good theta test sections'' motivated by the non-vanishing of theta lifts. In this paper, we aim to prove a partial analog for $\widetilde{{\rm Sp}_2}(\mathbb{Q}_2)$. However, due to several complexities, we compute the doubling integral only for certain irreducible principal series representations induced from characters with ramified quadratic twists. We develop some $2$-adic analogs for the machinery in the paper mentioned above; however, these tend to be more delicate and have more restrictive hypotheses than the non-dyadic case. Ultimately, this paper and the one mentioned above develop several computations intended to be used for future research into the non-vanishing of theta lifts.