Transfer of Wakamatsu tilting modules along Frobenius extensions

Let $ι: R\to A$ be a Frobenius extension and let $T$ be a Wakamatsu tilting left $R$-module. We give sufficient conditions for the induced module $A\otimes_R T$ to remain Wakamatsu tilting and establish an ascent--descent result for split centrally projective Frobenius extensions. We also characterize the tilting case by the vanishing of $\mathrm{Ext}_R^i(T,A\otimes_R T)$ for all $i>0$. Moreover, if $A\otimes_R T\in\mathrm{add}_R(T)$, then the natural map $S=\mathrm{End}_R(T)\to B=\mathrm{End}_A(A\otimes_R T)$ of endomorphism rings is a Frobenius extension and $T\otimes_S B\cong A\otimes_R T$ as $R$-$B$-bimodules. Applications to Brenner--Butler--Miyashita equivalences and some specific classes of Frobenius extensions are also discussed.